j^jLTU^nrm  u  JKT"5t  L  (T  E  B  R  A 

O  1.1  A  L    I ;  X  E  R  ^  '  s  r> 


VLGEBRA; 


COMMON    SCHOOLS. 


Bv  DAVID  B.  TOWER,  A.M., 

ite  Prinoipal  of  the  Eliot  Grannmar  School,  Bosfcob.  and  of  >:h 

Penn.  lusfcit.uta  f'or.Sb'e  Instrvict-ion  ct  the  Blind  ;  au'Aiof  or 

"  The  Gradual  Reader,  or  ExeTcises  in  AvhoulatiL  j." 


pur    "SHED  BY  DivNIM     •'       (^    S     S^  CO., 

(^L  ATE     OA  'J  Y     A   -'  I'     ■'■  ' 
60  JOIIN-STKKi:  ! 

awe."  .'racBC  STsa:  JTSKI . 


IN   MEMORIAM 
FLORIAN  CAJORl 


INTELLECTUAL    ALGEBRA; 

OR, 

ORAL      EXERCISES 

IN 

•  ALGEBRA; 

FOR 

COMMON     SCHOOLS, 

IN   WHICH 

ALL   THE   OPERATIONS   ARE    LIMITED   TO  SUCH  SMALL    NUMBERS   AS 

NOT   TO  EMBARRASS   THE    REASONING   POWERS,   BUT,   ON   THE 

INDUCTIVE  PLAN,  TO  LEAD  THE  PUPIL  UNDERSTANDINGLY, 

STEP  BY   STEP,   TO    HIGHER  MENTAL    EFFORTS: 


TO   PREPARE   THE   PUPIL   FOR  THE   STUDY    OP   WRITTEN 
ARITHMETIC, 

AND    DESIGNED    TO    Bf. 

IIVTRODUCTORY  TO   HIGHER  TREATISES  ON  ALGEBRA 


DAVID    B.  TOWEK,   A.  M., 

I  ATE    PRINCIPAL    OF    THE    ELIOT    GRAMMAR    SCHOOL,    BOSTON,    AND    OF    TH« 

PKNN.    INSTITUTE    FOR    THE    INSTRUCTION    OF    THE    BLIND  j    AUTH4  R 

OF  "the  GRADUAL  READER,  OR  EXERCISES   IN  ARTICULATION." 

"Divide  and  subdivide  a  diiTiniK  procew  until  your  eteps  are  so  sliort  ihat  llie 
pupil  can  easily  taltc  Ihem."  — Abbott's  Teacher. 

THIRTEENTH    EDITION. 

NEW  YORK: 
PUBLISHED  BY  DANIEL  BURGESS  &  CO., 

60  JOHN-STREET. 

1S55. 


Entered  according  to  Act  of  Congress,  in  the  year  1845,  by 

David  B.  Toweu. 
In  tne  Clerk's  Office  of  the  District  Court  of  Massachusetts. 


PREEACE. 


It  is  now  three  years  and  a  half  since  this  work  was  pre- 
pared for  the  use  of  the  blind  under  the  author's  charge  ; 
and  it  took  this  form  from  the  necessity  for  oral  instruction, 
in  their  peculiar  case.  The  great  advantages  derived  by 
them  from  these  exercises,  in  developing  and  strengthening 
the  mental  powers,  in  fixing  the  attention,  and  in  awakening 
a  strong  desire  to  acquire  knowledge  understandingly,  by 
seeking  the  icliy  and  wherefore  at  every  step  in  their  prog- 
ress, wrought  in  the  author  a  firm  conviction  that  algebra, 
in  this  shape,  should  precede  written  arithmetic;  and  that 
such  would  be  the  case  at  no  distant  period.  About  three 
years  ago,  the  design  of  the  author  to  publish  a  mental  alge- 
bra, was  communicated  to  two  of  the  most  distinguished 
teachers  in  this  city,  and  met  with  their  approval ;  but 
unceasing  duties  in  managing  a  large  institution,  have  hith- 
erto delayed  its  publication.  It  is  now  printed  for  the  use 
of  the  private  pupils  under  the  author's  care  ;  and,  with 
the  hope  that  it  may  be  as  successful  with  the  seeing,  as  it 
has  been  with  the  blind,  it  is  humbly  offered  to  the  public. 
Should  it  succeed  in  making  algebra  a  common  school 
study,  and  should  it  do  for  that  study,  in  some  small  de- 
gree, what  "  Colburn's  First  Lessons  "  have  done  for  arith- 
metic, the  author  will  congratulate  himself  that  one  original 
idea  of  his  has  been  of  value  to  the  young. 

These  exercises  gradually  lead  the  pupil,  step  by  step, 
from  the  simplest  to  more  complicated  reasoning  ;  teaching 
only  one   thing   at   a   time,  and   rendering   that    one    thing 


iw30f>i  .''^8 


faviiliar,  before  the  attention  is  called  to  another.  The 
additional  strength  that  the  mind  daily  gains  by  such  re- 
peated exercise,  can  hardly  be  conceived  but  by  the  ex- 
perienced teacher.  The  increase  of  intellectual  power  from 
such  a  source,  almost  equals  the  accession  of  physicaj 
strength,  which  ancient  fable  tells  us  a  man  acquired,  by 
carrying  a  calf  daily  till  it  grew  to  be  an  ox. 

Nor  will  an  algebraic  process  of  reasoning,  however  long, 
seem  at  all  difficult  for  the  memory,  where  the  numbers 
are  small,  when  it  is  recollected  that  you  have  to  stand  on 
but  one  round  of  a  ladder  to  reach  the  next  higher  round  ; 
and  that  this  process,  continued,  easily  caril?s  you  to  the 
top.  The  last  step  requires  no  greater  effort  than  the  first. 
So  in  an  algebraic  solution,  where  only  one  symbol  is  used 
to  express  the  conditions  of  a  question,  one  step  only  need 
be  held  in  mind  to  reach  the  next,  and  it  need  be  held 
only  till  the  next  is  reached.  Each  successive  step  is  de 
pendent  on  the  preceding,  and  is  derived  from  it  by  a 
process  of  reasoning  generally  limited  to  that  step.  Even 
in  using  several  symbols,  the  mind  is  easily  trained  to 
discriminate  and  retain  whatever  is  needed  as  an  argu- 
ment in  the  solution,  and  to  lay  aside  at  once  all  the  steps 
of  the  process  by  which  that  conclusion  was  reached. 

The  author  found  that  his  Wuid  pupils,  thus  taught, 
gained  intellectual  strength  sufficient  to  solve,  mentally, 
questions  requiring  five  diffijrent  letters  and  equations  to 
express  the  conditions.  That  part  of  the  manuscript,  how- 
ever, has  been  omitted  ;  ana  all  such  questions  have  been 
carcfull}'  excluded,  as  not  coming  within  the  design  of  this 
elementary  treatise. 

Furthermore,  an  algebraic  solution  is  far  less  mechanical 
than  an  arithmetical  one  is  often  permitted  to  be.  There 
is  no  remembering  abstract  numbers,  to  undergo  operations 
prescribed  by  rule ;  but  the  reasoning  on  each  successive 
step  attaches  a  meaning  to  it,  dependent  on  the  connection 


TREFACE.  O 

between  the  several  parts  of  an  equation.  Thus,  the  pupil 
is  delighted  to  exercise  his  powers  on  an  equation ;  it  is  a 
conflict  which  excites  his  mental  energy ;  and  who  remem- 
bers not  his  boyish  satisfaction  in  surmounting  a  diffi- 
culty ?  There  is  a  peculiar  pleasure  in  this  study,  when 
rightly  presented  to  the  young,  which  seldom  fails  to  in- 
terest and  rouse  the  pupil,  though  no  other  study  has  been 
able  to  call  forth  any  vigorous  effort.  Curiosity,  in  youth, 
the  main  spring  of  intellect,  is  hereby  made  to  act  in  its 
proper  sphere  ;  the  kind  interest,  the  skill,  and'the  superior 
intelligence  of  the  teacher,  must  direct,  while  this  curiosity 
needs  a  guide  ;  but,  once  on  the  track,  with  such  a  motive 
power,  the  wheels  can  never  cease  to  revolve. 

Every  teacher  knows,  from  experience,  how  readily  a 
pupil  will  understand  an  arithmetical  question,  and  with 
what  facility  he  will  reason  upon  it,  when  small  numbers 
are  substituted  for  large  ones,  without  altering  a  single 
condition  of  the  question,  however  difficult  and  unintelli- 
gible it  appeared  before.  Large  numbers  embarrass  the 
pupil ;  and  he  should  learn  to  reason  with  small  numbers 
at  first,  till  he  gradually  jKiquires  strength  to  wield  larger 
ones.  On  this  principle  the  author  has  based  this  work. 
The  numbers  are  small,  that  the  pupil  may  solve  the  ques- 
tions vicntally.  Although  intended  solely  as  oral  exercises, 
the  teacher  will  perceive  that  the  questions  may  be  solved 
on  the  slate,  and  that  written  algebra  can  be  taught  from 
this  hook  as  well  as  from  a  larger  treatise. 

A  Key,  containing  answers,  solutions,  and  suggestions  for 
teachers,  is  in  press,  and  will  be  of  assistance,  especially  to 
those  who  have  neither  taught  nor  studied  algebra. 

The  author  invariably  required  his  pupils  to  make  ques- 
tions in  each  successive  section ;  thus  he  ascertained  that 
each  principle  was  clearly  understood  before  he  proceeded 
to  the  next.  This  would  be  found  very  useful,  and  might 
be  made  a  home  excrci.se. 


O  PREFACE. 

To  his  brethren  the  Boston  Teachers  this  work  is  re 
epectfully  dedicated,  by  their  friend  and  former  associate, 
with  the  earnest  desire  that  their  efforts  in  the  cause  of 
education,  and  their  devotion  to  the  interests  of  the  younf 
may  be  duly  appreciated  and  rewarded 

D.    li.    >* 

Ko.  13,  Somerset  Steket, 
Boston,  April  8,  iK\c>. 


SUGGESTIONS  TO  TEACHERS. 


ArrER  a  question  is  read  or  given  out,  cail  on  some 
cne  of  the  class  to  repeat  it ;  on  another,  to  state  what 
is  required  ;  on  a  third,  for  the  data,  or  known  con- 
ditions on  which  the  question  is  based,  and  from 
which  the  answer  is  to  be  deduced;  on  a  fourth,  to 
state  what  x,  or  any  other  symbol  vised,  is  to  represent 
in  the  given  case  ;  on  a  fifth,  to  use  a  symbol  or  sym- 
bols in  accordance  with  the  expressed  conditions  ;  on 
a  sixth,  to  make  an  equation  from  the  materials,  using 
the  symbols  to  represent  the  unknown  quantities  ;  on 
a  seventh,  to  prove  the  equation,  thus  made,  to  be  true, 
stating  why  and  wherefore;  on  an  eighth,  to  give  the 
first  step  in  reducing  this  equation,  with  the  reasons 
for  it ;  on  a  ninth,  for  the  7iext  step  ;  and  so  on,  till 
the  value  of  the  symbol  or  symbols  used,  is  found 
Another  pupil  should  then  be  required  to  prove  the 
whole,  by  using  the  numerical  value  thus  found  in  the 
several  conditions  of  the  question.  When  there  are 
several  ways  of  reducing  an  equation,  other  pupils  can 
go  through  with  each  of  thein  in  the  same  manner. 
By  calling  on  the  pupils  promiscuously,  the  attention 
of  all  is  thus  confined  to  each  step  of  ttie  process,  and 
the  greatest  benefit  is  secured.  By  this  method,  in  a 
class  of  forty,  each  pupil  does  something  in  the  way 
of  recitation,  towards  the  solution  of  every  third  or 
f<5urih  question,  and  silently  is  compelled  to  attend  to 
tht  whole  process  of  all. 


fi  INTELLECTUAL     ALGEBRA. 

Ill  addition  to  the  oral  lesson  of  the  day,  thus  re 
cited,  the  class  may  be  lequired  to  have,  on  thei: 
slates,  the  lesson  of  the  preceding  day.  Here,  too,  each 
step  of  the  solution  should  be  explauied  in  a  similar 
manner  by  the  class.  This  will  serve  for  a  review, 
and,  at  tiie  same  time,  teach  tvritten  Algebra. 

The  Key  will  be  found  of  great  assistance  to  the 
teacher  in  hearing  a  class,  interrupted,  as  he  constantly 
is,  by  the  many  who  demand  his  care  and  attention 
Besides,  it  will  b^  useful  for  assistants  and  monitors. 


n.j 


INTELLECTUAL  ALGEBRA. 


SECTION  I 


1.  In  one  scale  are  two  cannou  balls,  of  equaj 
weight;  in  the  other  scale  are  placed  one-pound 
weights  enough  to  balance  the  two  balls. 

Here  is  a  balancing  or  equality  of  weights.  And 
since  it  takes  six  one-pound  weights  to  balance  the 
two  balls,  two  balls  weigh  six  pounds ;  and  the  ex- 
pression, 

Ttco  balls  are  equal  to  six  pounds, 
is  an  equation.     This  equation  may,  by  using  =    ihe 
s\gn  of  equality,  be  expressed  thus : 
2  balls  =z  6  pounds. 


8  INTELLECTUAL     ALGEBRA.  [§   1. 

2.  In  the  equation 

2  balls  :=  6  pounds, 
/he  number  of  pounds  needed  to  balance  tine  ball,  or 
the  tpeiglit  of  one  ball,  is  the  unknown  quantity  to  be 
found  out  or  determined. 

If  six  pounds  balance  two  balls,  it  is  evident,  that 
one  half  as  many  pounds  will  balance  07ie  ball  ;  or, 
if  two  balls  weigh  six  pounds,  one  ball  will  weigh  one 
half  of  six  pounds,  which  is  three  pounds ;  because, 
if  one  ball  weighs  three  pounds,  two  balls  will  weigh 
two  times  three  pounds,  which  is  six  pounds. 

3.  In  algebra,  some  symbol,  as  the  letter  x,  or  y,  is 
used  to  represent  the  unknown  or  undetermined  num- 
ber; that  is,  the  thing,  or  things,  required  to  be  found. 

In  the  equation 

2  balls  :=■  6  pounds, 
if  the  weight  of  one  ball  is  required,  the  unknown 
quantity  or  thing  sought,  is  the  weight  of  one  ball. 
If  a  symbol,  as  the  letter  x,  is  used  for  the  weight  of 
one  ball,  the  unknown  quantity,  x,  will  represent  the 
number  of  pounds  that  one  ball  weighs. 

4.  If  2;  represents  the  number  of  pounds  that  one 
ball  weighs,  two  times  x  will  stand  for  the  number 
of  pounds  the  two  balls  weigh ;  that  is,  if  one  ball 
weighs  X  pounds,  two  balls  will  weigh  two  times  x 
pounds,  which  may  be  expressed  thus  :  2  x  jjounds. 
Since  two  balls  weigh  six  pounds,  2  x  pounds,  repre' 
senting  the  weight  of  two  balls,  must  be  equal  to  six 
pounds.     We  have,  then,  this  equation, 

2  x  pounds  ■=■  6  pounds, 
and  2x  is  one  member  of  the  equation,  and  6  is  the 


§   l.J  INTELLECTUAL     ALGEBRA.  9 

other  member.     Now,  we  wish  to  find  the  value  of  z, 
or  the  number  that  it  represents. 
.5.  In  the  equation 

2  X  pounds  =1  6  pounds, 
one  X,  which  is  one  half  of  two  x,  must  be  equal  to  o?ie 
half  of  six  pounds,  which  is  three  pounds.  Then 
three  is  the  number  represented  by  x,  and  the  value  of 
X  is  now  known  or  determined  to  be  three.  The  weight 
of  one  ball  is  therefore  three  pounds  ;  and  one  ball  in 
OTJe  scale  will  balance  three  pounds  in  the  other. 

6.  Since  one  ball  in  one  scale  balances  three  pounds 
in  the  other  ;  if  o«€  more  ball,  of  the  same  weight,  be 
put  into  the  scale  with  the  first  ball,  it  is  evident  that 
three  one-poimd  iceights  must  be  added  to  the  three 
pounds  already  in  the  other  scale,  that  the  balance  or 
equality  may  still  be  preserved.  If  a  third  ball  be 
put  with  the  two  balls,  three  more  pounds  must  be  put 
with  the  six  pounds,  that  the  balance  or  equation  of 
weight  may  still  exist. 

7.  Therefore,  if  equal  tffeights  be  added  to  each 
scale,  when  the  scales  are  balanced,  the  balance  or 
equality  continues.  We  also  see,  that  if  one  ball=^ 
three  pounds  in  weight,  two  times  one  ball,  that  is,  two 
hd\\s,^=two  times  three  pounds,  that  is,  six  pounds; 
and  three  times  one  ball,  or  three  balls,  ::=  three  times 
three  pounds,  or  nine  pounds  ;  so  that,  li  equal  weights 
be  equally  increased,  the  balance  or  equality  between 
them  still  exists. 

8.  In  the  equation 

a;  =  3, 
if  X  be  added  to  x,  the  first  member,  it  is  evident  that 
3,  the  value  of  x,  or  number  it  represents,  must  be 


10  INTELLECTUAL     ALGEBRA.  [§   1 

udded  to  3,  the  second  member,  that  the  equulity  may 
be  preserved.     Then 

z  added  to  x  will  be  equal  to  3  added  to  3; 

or,  X  and  x  =:  3  and  3  ; 

or,  using  plus,  -j-,  the  sign  of  addition 

X  +  X  =  3  +  3. 

But    x-|-xr=2x,    and   3-}-3:=G;     therefore,    the 

equation  now  is, 

2xzrr6. 
If  X  be  again  added  to  the  first  member,  and  the 
number  it  represents  be  added  again  to  the  second 
member,  the  equation  will  be 

x-f2x  =  3  +  6, 
or,  3x  =  9. 

9.  Therefore,  if  equal  quantities  be  added  to  each 
member  of  an  equation,  the  equality  still  continues. 
We  also  see  that  if  x  =  3, 

twice  X  r=  twice  3, 

■  and  using  X ,  the  sign  of  multiplication, 

2X  x  =  2X3; 

so  that,  if  each  member  of  an  equation  be  multiplied 

by  the  same  number,  the  equation  or  equality  will  still 

be  preserved. 

10.  Two  balls  in  one  scale  balance  six  pounds  in  the 
other,  and  each  ball  weighs  three  pounds.  If  one  of 
the  two  balls  be  taken  out  of  the  scale,  it  is  evident 
that  its  weight,  or  three  one-pound  weights,  must  be 
taken  from  the  other  scale,  that  the  balance  or 
equality  of  weights  may  still  continue.  Then  the 
equation  will  be,  two  balls  with  one  ball  taken  front* 
them  =z  G  pounds  with  3  pounds  taken  from  them. 

1 1 .  Therefore,  iT  equal  tocights  are  taken  from  each 


§   l.J  INTELLECTUAL     ALGEBRA.  IJ 

scale  when  the  scales  are  balanced,  the  balance  or 
equality  continues.  We  see  also,  that  if  two  balls  = 
six  pounds  in  weight,  one  half  of  two  balls,  which  is 
one  ball,  is  equal  in  weight  to  one  half  of  six  pounds, 
which  is  three  pounds  ;  or,  2  balls  divided  by  2  ::=  6 
pounds  divided  by  2 ;  that  is,  if  equal  weights  are 
equally  diminished,  or  if  the  same  part  of  equal 
weights  is  taken  away,  the  balance  or  equality  be- 
tween the  remaining  parts  still  exists. 

12.  If  X  be  taken  from  2x,  the  Jirst  member  of  the 
equation, 

2  a;  =  6, 
it  is  evident  that  3,  the  value  of  x,  or  the  number  that 
it  represents,  must  be  taken  from  6,  the  second  mem- 
ber, that  the  equality  may  be  preserved.  Then  2  x 
with  X  taken  from  them  ::=  6  with  3  taken  from  them  ; 
that  is, 

2  X  less  x=:G  less  3 ; 
or,  using  — ,  the  sign  of  subtraction,  called  minus, 
or  less, 

2a;  — x  =  6  — 3. 
But  2x  —  x  =  x;   and  6  —  3  =  3;    and  the  equation 
now  is, 

x  =  2. 

13.  Therefore,  if  equal  quantities  be  taken  from 

each  member  of  an  equation,  the  equality  still  continues 

between  the  remaining  parts  of  each  member.     We 

also  see,  that  if  2  a;  =:  6 ;  2  a;  divided  by  2,  that  is, 

...        2  a; 
using  a  sign  of  division,  —  =:  6  divided  by  2,  that  is, 

I ;   or  that  one  half  of  2a;=:one  half  of  6. 

So  that,  if  each  member  of  an  equation  be  divided 


12 


INTEL.LECTLAL     ALGEBRA. 


L§2. 


by  the  saine  number,  the  equality  or  equation  will  still 
be  preserved. 

Remark.  —  Sucli  questions  should  be  asked  as  arc  necessary 
to  ascertain  the  accuracy  and  clearness  of  the  pupil's  knowl- 
edge of  this  section.  This  each  teacher  will  do  for  himsell 
better  than  the  author  can  do  it  for  him.  Printed  questions, 
for  such  purposes,  in  the  hands  of  a  pupil,  too  often  serve  but 
as  moulds  in  which  to  run  his  answers ;  preventing,  rather 
than  aiding,  mental  effort. 


SECTION   II. 


'1.  If  two  cannon  balls,  of  equal  weight,  in  one 
wcale,  are  balanced  by  si.x  one-pound  weights  placed 
\n  the  other  scale,  how  many  of  these  one-pound 
weights  will  it  take  to  balance  one  ball  ?  or,  what  is 
the  weight  of  one  ball  f 


§2.]  INTELI.ECTUA1-     AIX.EBRA.  13 

Explanation  and  Solution. 

Let    %   represent    the    ansicer   sought,    or    unknown 

quantity ; 

that  is,  let  x  represent  the  weight  of  one  ball. 

Then,  two  times  x  pounds  will  stand  for  the  weight 

of  two  balls  ;  thus, 

two  balls  weigh  2  x  pounds. 

But,  by  a  statement  or  condition  of  the  question, 

the  tw'o  balls  weigh  six  pounds. 

Therefore,  2x  pounds  must  be  equal  to  six  pounds. 

If  2x  pounds  are  equal  to  six  pounds, 

or  2  a:  =  6  ; 

th(  n  one  x  will  be  equal  to  one  half  of  six  pounds , 

•  If,  one  half  of  2  x,  which  is  x,  is  equal  to  one  half 

of  six  pounds,  which  is  three  pounds. 

Or  it  may  be  expressed  thus  • 

x=z  ^■ 

that  is,  x  equals  six  divided  by  two. 

Answer.     The  weight  of  a  ball  is  3  lbs. 

2.  One  X  is  what  part  of  two  times  x? 

3.  In  the  equation  2  2:  =  8,  to  what  part  of  eight  is 
one  X  equal  ? 

4.  If  each  member  of  the  equation  2  x=z8,  be 
divided  by  two,  what  equation  will  express  the  quo- 
tient ? 

5.  What  will  represent  the  sum  of  x  and  x  ? 

6.  Express  in  one  term  the  three  terms  x-\-x-\-x. 
What  will  represent  their  sum  ? 

7.  If  each  member  of  the  equation  3  3:=  12,  be 
divided  by  3,  what  equation  will  express  the  re- 
sult? 


14  INTELLECTUAL     ALGEBRA.  [§  2 

8.  George  and  Charles  are  to  have  four  balls ;  and 
one  is  to  have  as  many  as  the  other.  How  many  balls 
will  each  have  ? 

Let  I  represent  the  number  of  balls  that  George  will 

have. 
Then  x  will   also  represent  the  number  Charles  will 

have. 

Now,  if  r,  or  George's  number  of  balls,  be  added  to 

X,  or  Charles's  number  of   balls,  the  sum  will  be 

X  -|-  a;  r=  2  z  balls. 

2x  will  represent  the  number  of  balls  that  both  will 

have ; 

and,  since  both  together  will  have  4  balls, 

2  x  balls  must  be  equal  to  4  balls. 

If  2x=r4  balls,  one  x  will  equal  one  half  of  4  balls, 

which  is  2  balls. 
Therefore,  x  z=:  2  balls,  and  each  boy  will  have  2  balls. 

9.  If  four  be  divided  into  two  equal  parts,  what  will 
one  of  the  parts  be  ? 

Let  xz=z  one  part; 

then  another  %  will  represent  the  other  part  ; 

and  x-\-x,  which  is  2  x,  will  represent  both  parts,  or 

the  whole  number. 

But  the  whole  number  is  4  ; 

therefore  2x^4. 

If  2  a:  =  4,  a;  will  equal  one  half  of  4. 

Therefore,  a;  z=  2, 

and  one  of  the  parts  is  2. 

10.  Mary  ;md  Anna,  together,  have  eight  books; 
and  Mary  has  as  many  as  Anna.  How  many  books 
has  each  ? 

n.  John   and  James  are  to  have  ecjual  shares  of 


^  2.]  INTELLECTUAL     ALGEBRA.  13 

eighteen  chestnuts.     How  many  chestnuts  will   each 
have  1 

12.  Two  boys  agree  to  take  equal  shares  of  twenty- 
four  apples.     How  many  apples  may  each  take  ? 

13.  What  number  must  be  added  to  itself,  that  the 
sum  may  be  six? 

14.  If  X  represents  some  number,  what  will  repre- 
sent the  same  number  added  to  itself? . 

15.  Robert  has.  sixteen  apples,  which  he  wishes  to 
divide  equally  between  John  and  himself  How  many 
apples  will  John  have  ? 

16.  When  a  certain  number  is  added  to  itself,  the 
sum  is  ten.  If  x  represents  the  number,  what  will 
represent  the  number  added  to  itself?  To  what  will 
the  number  added  to  itself  be  equal  ?  What  is  the 
number  ? 

17.  In  two  classes  there  are  thirty  pupils  ;  and  there 
IS  the  same  number  of  pupils  in  each  class.  If  x 
represents  the  number  in  one  class,  what  expression 
will  represent  the  number  in  both  classes  ?  To  what 
must  this  expression,  that  represents  the  number  in 
both,  be  equal  ?  What  is  the  number  of  pupils  in 
each  class  ? 

18.  What  number  must  be  added  to  itself,  that  the 
sum  may  be  twenty-four  ? 

19.  Add  such  a  number  to  itself,  that  the  sum  shall 
be  sixteen.  What  will  represent  the  number  added  to 
itself?  To  what  will  the  number  added  to  itself  be 
equal  ?     What  will  the  number  be  ? 

20.  Add  such  a  number  to  itself  that  the  sum  shall 
be  thirty.     What  will  the  number  be  ? 

21.  Two  boj's  together  have  twenty-two  cents  ;  and 


16  liNTKLLKCTl  AL      ALGEBRA.  '  5  -^ 

one  has  as  many  as  the  other.  If  x  represents  the 
number  of  cents  that  one  boy  has,  what  will  represent 
the  number  that  both  have?  To  what  will  this  ex- 
pression be  equal?     How  majiy  cents  will  each  have? 

22.  What  number  must  be  added  to  itself,  that  the 
sum  may  be  twenty  ? 

23.  There  are  eighteen  chairs  standing  in  two  rows, 
with  the  same  number  in  each  row.  How  many  chairs 
are  there  in  each  row  ? 

24.  Divide  12  into  two  equal  parts.  If  x  represents 
one  of  the  parts,  what  will  represent  the  other  ? 
What  will  represent  both  parts?  What  will  both 
parts  equal  ?     What  will  one  part  be  ? 

25.  George  is  as  old  as  John,  and  the  sum  of  their 
ages  is  twenty-six  years.  If  x  be  used  to  represent 
the  age  of  John,  what  will  represent  George's  age  ? 
What  expression  will  stand  for  the  sum  of  their  ages  ? 
What  will  this  expression  equal?  What  is  the  age  of 
each  ? 

26.  In  the  equation  2  a:  =:  28  cents,  what  is  the 
value  of  X  1 

27.  Anna  gave  some  money  to  a  poor  woman,  and 
Josephine  gave  her  as  much  as  Anna  did.  She  re- 
ceived from  both  thirty  cents.  How  many  cents  did 
each  give  her  ? 

28.  If  a  line,  fifty  feet  long,  be  cut  into  two  equal 
parts,  how  long  will  one  of  the  parts  be  ? 

29.  What  number  must  be  added  to  itself,  that  the 
8  im  may  be  sixty  ? 

30.  The  number  of  full  barrels  in  a  store  is  equal 
o  the  number  of  empty  ones ;  and  the  sum  of  both 
3  forty.     How  many  are  there  of  each  kind  ? 


§2.]  INTELLECTUAL     ALGEBRA.  11 

3J.  In  a  school  of  forty  pupils,  there  are  as  many 
boys  as  girls.    How  many  pupils  are  there  of  each  sex  ? 

32.  Twenty-four  horses  and  cows  are  feeding  m  a 
pasture,  of  each  an  equal  number.  How  many  are 
there  of  each  ? 

33.  If  a  line,  thirty-three  feet  long,  be  cut  into  three 
equal  pieces,  and  if  x  represents  one  of  the  pieces, 
what  will  represent  each  of  the  other  two  pieces  1 
What  will  represent  the  sum  of  the  pieces  ?  To  what 
will  the  expression,  that  represents  the  sum  of  the 
pieces,  be  equal  ?    How  long  will  each  of  the  pieces  be  1 

34.  John,  Charles,  and  Caleb,  have  each  an  equal 
number  of  blocks,  and  together  they  have  twenty-one. 
How  many  blocks  has  each  ? 

35.  Divide  fifteen  into  three  equal  parts.  If  x  rep- 
resents one  of  the  parts,  what  will  stand  for  each  of 
the  other  two  parts  1  What  will  express  the  sum  of 
the  parts  ?     What  will  one  of  the  parts  be  ? 

36.  A  farmer  wishes  to  put  ninety  sheep  into  three 
pastures,  so  that  there  may  be  an  equal  number  in 
each.     How  many  sheep  will  there  be  in  each  pasture  ? 

37.  George,  Anna,  and  Charles,  are  to  have  equal 
shares  of  twenty-seven  peaches.  If  x  represents  An- 
na's share,  what  will  represent  the  share  of  each  of 
the  other  two  ?  What  will  represent  the  sum  of  tbeir 
shares?  What-will  the  expression  for  the  sum  equal? 
How  many  peaches  will  each  have  ? 

38.  Divide  thirty-six  plums  among  three  boys, 
giving  the  same  number  to  each.  How  many  plums 
will  each  boy  receive  ? 

39.  What  number  must  be  added  once  to  itself 
that  the  sum  may  be  forty  1 


18  INTELLECTUAL     ALGEBRA.  [§  3, 

40.  Wliat  number  must  be  added  twice  to  itself 
that  the  sum  may  be  eighteen  ? 

41.  Four  men  contributed  equally  to  purchase  <» 
cow  for  a  poor  jieighbor.  The  cow  cost  twenty-four 
dollars.     IIow  many  dollars  did  each  man  give? 

42.  What  number  must  be  added  twice  to  itself, 
hat  the  sum  may  be  thirty  ? 

43.  Divide  thirty-six  into  four  equal  parts.  If  z 
equals  one  part,  what  will  be  the  sum  of  the  parts? 
Of  how  many  will  one  of  the  parts  consist  ? 

44.  What  number  must  be  added  three  times  to 
itself,  that  the  sum  may  be  twenty-four  ? 

45.  If  4  i-  =  20,  what  part  of  20  will  one  z  equal  ? 

46.  If  a  number  be  added  four  times  to  itself,  the 
sum  will  be  thirty-five.     What  is  the  number  ? 

47.  W'hat  will  express  the  sum  of  x-\-x-\-x-\-x? 

48.  If  X,  and  x,  and  x,  be  added,  what  will  be  the 
sum  ? 

49.  Express  in  one  term,  the  sum  of  a; -|- a^ -j"  ^  ~h 
x-\-x. 

50.  How  many  times  x  are  x-\-x-\-x  ?  What 
term  will  express  the  sum  ? 


SECTION   III. 

1.  Two  cannon  balls,  one  weighing  twice  as  much 
as  the  other,  placed  in  one  scale,  are  balanced  by 
twelve  ore-pound  weights  m  the  other  scale.  What 
is  the  weight  of  each  ball  ? 


§3.]  INTELLECTUAL     ALGEBRA.  19 

Let  a;r=the  weight  of  the  lighter  ball  ; 

ihen  x-\-x=.2z  will  be  the  weight  of  the  other  ball, 

and  z-|~2^  =  3a;  will  represent  the  weight  of  both 

balls. 

Both  balls,  then,  will  weigh  3  x  pounds. 

But,  by   a  condition  of  the  question,  the  two  balls 

weigh  12  pounds. 

Therefore,  3  x  pounds  must  be  equal  to  12  pounds ; 

expressed  thus  ;  3  xz=:  12. 

If  Sz  pounds  are  equal  to  12  pounds, 

X  pounds,  which  is  one  third  of  3  z  pounds, 

•  will  be  equal  to  one  third  of  12  pounds ; 

therefore,   z  =  4   pounds,  the   weight  of  the   lighter 

ball ; 

and  2x=z  twice  4,  which  is  8  pounds  for  the  heavier 

ball. 

Or, 

dividing  each  member  of  the  equation, 

3xrrl2, 

by  3,  gives  the  new  equation 

z:=4,  as  above. 

2.  If  3  z=:  12,  to  what  part  of  12  will  x  be  equal  ? 

3.  If  X  be  added  to  2  x,  what  term  will  express  their 
sum  1 

4.  If  each  member  of  the  equation  5xz=  15,  be 
divided  by  5,  what  equation  will  represent  the  quo- 
tient ? 

5.  If  x-\-2x-\-3x  be  united  in  one  term,  what 
will  represent  their  sum  ? 

6.  Divide  nine  balls  between  George  and  Charles, 
giving  to  George  twice  as  many  as  to  Charles.  How 
many  balls  will  each  receive  ? 


20  INTELLECTUAL     ALGEBRA.  [§  •^• 

Let  .T  represent  the  undetermined  or  unknown  number 

of  balls  which  Charles  is  to  have ; 
then  two  times  x,  or  twice  the  unknown  number,  will 
stand  for  the  number  of  balls  which  George  is  to 
have. 
■Vow,  T,  or  Charles's  share,  added  to  2  x,  or  George's 
share,  will  be  3t,  or  the  number  of  balls  that  both 
will  have. 

But  both  together  are  to  have  9  balls  ; 

therefore,  3  x  balls  must  be  equal  to  9  balls; 

and  X  balls  will  be  equal  to  one  third  of  9  balls  ; 

then  a;  rr:  3  balls,  or  Charles's  share. 

Kince  X,  the  unknown  number,  is  found  to  represent  3 

balls,  2  T,  or  twice  the  number,  will  be  twice  3  balls : 

therefore,  George's  share  will  be  6  balls 

Or, 

dividing  the  equation 

3x  =  9 

by  3,  gives  the  new  equation 

X  :=  3,  as  above. 

/     In  the  equation   x -j- 2  a;  =  24,  what  number   ia 

represented  by  a;  ? 

8.  There  are  two  numbers,  one  of  which  is  twice 
the  other,  and  their  sum  is  fifteen.  If  x  represents 
the  smaller  number,  what  will  represent  the  larger? 
What  will  express  the  sum  of  the  two  numbers?  To 
what  will  the  sum  of  the  two  numbers  be  equal  ? 
What  are  the  numbers  ? 

9.  Anna  is  twice  as  old  as  Mary,  and  the  sum  of 
their  ages  is  eighteen  years.  What  is  the  age  of 
each  ? 

10.  The  sum  of  two  numbers  is  twelve,  and  one 
is  twice  the  other.     What  are  the  numbers  ? 


^^  3.1  INTELLECTUAL     ALGEBRA.  21 

11.  George  has  twice  as  many  books  as  Thomas, 
and  they  both  together  have  twenty-one.  How  many 
books  has  each  ? 

12.  Divide  eighteen  into  two  such  parts,  that  one 
part  shall  be  twice  the  other.  K  xz=  the  smaller 
part,  what  will  represent  the  larger  1  What  will  ex- 
press the  sum  of  the  parts  ?     What  will  the  parts  be  1 

13.  Charles  and  Anna  have  fifteen  blocks  for  play, 
but  Charles  has  twice  as  many  as  Anna.  How  many 
blocks  has  each  ? 

14  Divide  twenty-one  into  two  such  parts,  that 
one  part  shall  be  twice  the  other.  What  are  the 
parts  ? 

15.  Rollo  and  Lucy  picked  up  thirty  shells  on  a 
beach,  and  they  wish  to  divide  them  so  that  Lucy 
shall  have  twice  as  many  as  Rollo.  How  many  shells 
can  each  have? 

16.  There  are  two  numbers,  one  of  which  is  twice 
the  other,  and  their  sum  is  forty-five.  What  are  the 
numbers  ? 

17.  Divide  thirty-six  into  two  such  parts,  that  one 
shall  be  twice  the  other.     What  will  each  part  be  ? 

18.  In  a  pasture  there  are  twenty-four  horses  and 
cows  feeding  together,  and  the  number  of  cows  is 
double  the  number  of  horses.  How  many  are  there 
of  each 

19.  The  sum  of  two  numbers  is  thirty-nine,  and 
one  is  as  large  again  as  the  other.  What  are  the 
numbers  ? 

20.  What  number  must  be  added  twice  to  itself, 
ihat  the  sum  may  be  nine  1 

21     Find  two  numbers,  whose  sum  shall  be  forty- 


82  INTELLECTUAL     ALGEBRA.  [§  3. 

eight,  and  one  shall  be  twice  the  other.     What  vvilJ 
the  numbers  be  ? 

22.  AVhat  number  must  be  added  to  twice  itself, 
that  the  sum  may  be  thirty? 

23.  There  are  two  numbers,  one  of  which  is  three 
times  the  other,  and  their  sum  is  twenty-four.     If  x 
represents  the  smaller  number,  what  will  be  the  larger  ? 
What  will  express  their  sum  ?     What  are  the  num 
bers? 

24.  John  and  William  wish  to  divide  thirty-six 
cents,  so  that  John  shall  have  three  times  as  many  as 
William.     How  many  will  each  have  ? 

25.  What  number  must  be  added  twice  to  itself 
that  the  sum  may  be  twenty-one  ? 

2fi.  What  number  must  be  added  to  twice  itself 
that  the  sum  may  be  forty-eight  ? 

27.  The  sum  produced  by  adding  a  number  twice 
to  itself  is  twenty-seven.     What  is  the  number? 

28.  What  number  must  be  added  three  times  to 
itself,  that  the  sum  may  be  thirty-two  ? 

29.  Robert  and  Mary  wish  to  share  forty  peaches, 
so  that  Mary  may  have  three  times  as  many  as  Robert. 
How  many  can  each  have  ? 

30.  The  sum  of  two  numbers  is  sixty,  and  one  is 
four  times  the  other.     What  are  the  numbers  ? 

31.  If  thirty-six  be  divided  into  three  equal  partrf, 
what  will  one  of  the  parts  be  ? 

32.  What  number  must  be  added  four  times  to 
itself,  that  the  sum  may  be  thirty-five  ? 

33.  The  number  of  cows  and  sheep  together  in  a 
farm-yard  is  seventy-five,  and  there  are  four  times  as 
many  sheep  as  cows.     How  many  are  there  of  each  ? 


^  3.J  INTELLECTUAL     ALGEBRA.  23 

34.  Find  two  numbers  whose  sum  will  be  seventy- 
two,  and  one  will  be  five  times  the  other.  What  are 
the  numbers  ? 

So.  Divide  eighty-four  into  two  such  parts,  that  one 
shall  be  six  times  the  other.  What  will  one  of  the 
parts  be? 

36.  There  are  seven  times  as  many  sheep  as  lambs 
in  a  pasture,  and  in  all  there  are  ninety-six.  How 
many  are  there  of  each  ? 

37.  There  are  two  numbers,  one  of  which  is  eight 
times  the  other,  and  their  sum  is  fifty-four.  What 
are  the  numbers  1 

38.  What  number  must  be  added  to  nine  times  it- 
jelf,  that  the  sum  m.ay  be  one  hundred  and  thirty  ? 

39.  A  man  and  a  boy  together  receive  for  theli" 
work  fifty-five  dollars ;  and  the  man's  share  is  ten 
times  as  large  as  the  boy's.  How  many  dollars  does 
each  receive  ? 

40.  If  X  be  multiplied  by  3,  what  expression  will 
represent  the  product  ? 

41.  If  X  be  multiplied  by  7,  what  will  express  the 
product? 

42.  If  7  be  multiplied  by  x,  what  will  express  the 
product? 

43.  What  is  the  product  of  9  multiplied  by  a;  ? 
4-1.    What  is  the  product  of  x  multiplied  by  9  ? 

45.  What  will  represent  the  product  of  x  times  5  ? 

46.  What  expression  will  represent  x  times  12  ? 

47.  If  3  a;  be  added  to  4  x,  what  will  represent  their 
sum? 

48.  What  will  express  the  sum  of  2  x  -\- 3  x  -\-  4  xl 


24  INTELLECTUAI.     ALGEBRA.  §      3.  J 

49.  What  one  term  will  represent  the  sum  of 
5  z  +  3  a;  -f-  X  ? 

50.  Two  men,  working  for  a  dollar  a  day,  receive 
one  hundred  and  thirty-two  dollajs  for  their  labor. 
A  worked  eleven  days  to  B's  one.  If  x  represents  the 
number  of  days  B  worked,  what  Avill  represent  the 
days  A  worked  ?  What  will  express  the  number  of 
days  both  worked?  If  each  received  1  dollar  for  a 
day's  work,  what  will  express  the  number  of  dollars 
he  received  for  x  days?  \i  A  received  x  dollars  for 
working  x  days,  what  will  express  the  number  of 
dollars  he  received  for  working  11  x  days?  What  ex- 
pression will  represent  the  dollars  both  received,  and 
to  what  number  of  dollars  must  this  expression  be 
equal  ?  How  many  days  did  each  work,  and  how 
many  dollars  did  each  receive  ? 

51.  The  sum  of  two  numbers- is  two  hundred  ;  and 
one  is  nineteen  times  the  other.  If  x  represents  the 
smaller,  what  will  stand  for  the  larger  ?  What  are 
the  numbers? 

52.  One  of  two  numbers  is  twelve  times  the  other, 
and  their  sum  is  thirty-nine.     What  are  the  numbers  ? 

53.  Divide  one  hundred  and  fifty  dollars  between 
two  men,  giving  one  fourteen  nmes  as  many  as  the 
other.     How  many  dollars  will  each  receive  ? 

54.  George  and  Charles  paid  eighty  cents  for  a 
sled  ;  but  George  paid  fifteen  times  as  much  as 
Charles.     What  did  each  pay  ? 

55.  A  man  paid  one  hundred  and  twenty  dollars 
for  a  horse  and  saddle,  and  the  horse  cost  nine  times 
as  much  a'  ,he  saddle.  How  many  dollars  did  he 
pay  for  each  ' 


Cj  4. J  INTELLECTUAL      ALGEBRA.  25 

56.  What  number  must  be  added  to  nineteen  times 
itself,   that  the  sum  may  be  one  hundred  ? 

57.  Find  the  sum  of  3  x  -|-  (3  x  -[-  1'2  x. 

58.  What  is  the  sum  of  z  -f  2  z  +  4  a;  ? 

59.  Unite  the  following  terms  ;  x  -\-  Sz-\-9  x. 

60.  Express  in  one  term  the  sum  of  2x  -\-8x  -j-  x 

61 .  What  is  the  sum  of  G  x  -^  IQ  x  -{-  15  x  1 

62.  Three  to  be  multiplied  by  x,  by  using  X,  the 
sign  of  multiplication,  may  be  expressed  thus;  3  X  2; 
What  will  the  expression  be  after  the  multiplication  is 
performed  "? 


SECTION    IV. 

I.  One  cannon  ball  and  two  one-pound  weights  m 
one  scale,  are  balanced  by  six  one-pound  weights  in 
the  other  scale.     What  is  the  weight  of  the  ball  ? 

Let  X  represent  the  weight  of  the  ball. 
Then  x  pounds  plus  2  pounds  will  be  the  number  of 

pounds  in  one  scale. 
Since  these  balance  the  six  pounds  in  the  other  scale, 
X  pounds  and  2  pounds  are  equal  in  Vv^eight 
to  6  pounds  ;  that  is, 
x-\-2  =  6. 
Now,  if  2  pounds  be  taken  out  of  the  scale  contain- 
ing  the   ball  and   2  pounds,  it  is   evident   that  2 
pounds  must  be  taken  out  of  the  scale  containing 
the  6  one-pound  weights,  that  the  balance  or  equal' 
ity  may  be  preserved. 


26  INXELbECTCAI.     ALGEBRA.  [§  4. 

Thus  the  ball  alone  in  one  scale  balances  the  4  pounds 

ni  the  other. 

If  2  be  taken  from  each  member  of  the  equation, 

a;  -f  2  =z  C, 

the  equation  will  be,  x  -|-  2  —  2  =  0  —  2,  or  z  =  1. 

Therefore  the  weight  of  the  ball  is  4  lbs. 

2.  If  2  be  taken  from  the  expression  x  -\-  2,  whal 
will  be  the  remainder  ?  If  2  be  taken  from  0,  what 
will  be  the  remainder  ? 

3.  If  2  be  taken  from  each  member  of  the  equa- 
tion a;  -j-  2  =:  6,  what  equation  will  express  tiie  re- 
mainder ? 

4.  If  5  be  taken  from  each  member  of  the  equation 
x-\-Sz=  13,  what  e(}u.-Uion  will  reprei--ent  the  remain- 
der? 

5.  George  gave  two  apples  to  Charles,  and  then 
Charles  had  seven.  IIow  many  had  Charles  before 
he  received  the  two  from  George  ? 

(3.  If  4  ce  taken  from  each  member  of  the  equa- 
tion a;  -[-  4  rn  9,  the  subtraction  may  be  expressed 
thus  ;  a;  -{-  4  — 4  r=  9  —  4.  What  will  the  equati<m  be, 
after  the  terms  are  united  ? 

7.  Anna  says,  "  If  you  give  me  three  more  books, 
I  shall  have  fiReen."     How  many  has  she  now  ? 

8.  A  purse  lacks  five  cents  of  being  filled,  and  il 
will  hold  twenty  cents.     IIow  many  cents  are  in  it.? 

9.  A  room  contains  Iwent^'-four  chairs,  an  !  only 
six  are  empty.     How  many  are  occupied  ? 

10.  What  number  is  that,  to  which  if  six  be  added 
the  sum  v.'ill  be  fifteen  ? 

\\     Tlie    sum    of   two    numbers    is    ten,    and    thi 


^  4.]  INTELLECTUAI.     ALGKBRA.  27 

larger  number   is  two  more  than  the  smaller.     What 
are  the  numbers  ? 

Let  X  represent  the  smaller  number  ; 

then  x-\-'i  will  be  the  larger  ; 

and  x-j-z-f-2,  or  2  a; -{-2,  will   be   the  sum  of  the 

numbers. 

Therefore,  2  x-|- 2  =  10. 

Subtracting  2  from    each  member    of  the  equation, 

gives 

2x-j-2  — 2=10  — 2; 

uniting  terms,  2  x  =  8. 

Dividing  each  member  of  the  equation  by  2,  gives 

a;:=f=4,  the  smaller; 

then  X  -}-2  =:  6,  the  greater. 

12.  The  larger  of  two  numbers  is  five  more  than 
the  smaller,  and  their  sum  is  seventeen.  What  are 
the  numbers  1 

13.  There  are  eight  books  on  a  shelf;  a  part  be- 
longing to  Mary,  and  the  remainder  to  Anna ;  but 
Mary  owns  two  more  than  Anna.  How  many  belong 
to  each  I 

14.  Divide  twenty-three  chestnuts  between  John 
and  James,  giving  to  James  five  more  than  to  John. 
How  many  will  each  have  ? 

15.  "  Father,"  says  Thomas,  "  if  I  had  ten  cents 
more,  I  could  buy  a  new  sled ;  but  the  carpenter  will 
not  make  me  one  for  less  than  a  dollar."  How  many 
cents  has  Thomas  ? 

16.  A  railroad  car  has  seats  for  sixty  persons,  and 
all  the  seats,  except  eight,  are  filled.  How  many 
passengers  are  in  the  car  \ 

17.  George  and  Charles  together  hrid  twenty-two 


28  INTELLECTUAL     ALGEBRA.  [§  4. 

cents,  but  George  had  four  more  than  Cliarlcs.     How 
many  had  each  ? 

18.  The  sum  of  two  numbers  is  twenty,  and  their 
difierence  is  six.     What  are  tlie  numbers  1 

19.  James  travels  nine  miles  more  than  William, 
and  the  distance  both  travel  is  forty-nine  miles.  How 
far  does  each  travel  1 

20.  The  sum  of  two  numbers  is  twenty-"five,  and 
the  larger  is  seven  more  than  the  smaller.  What  are 
the  numbers  1 

21.  Put  forty-one  apples  into  two  baskets,  so  that 
one  basket  shall  contain  eleven  more  than  the  other. 
How  many  will  you  put  into  each  basket  ? 

22.  There  are  two  numbers,  one  of  which  is 
t^velve  greater  than  the  other,  and  their  sum  is  thirty- 
two.     What  are  the  numbers  ? 

23.  Divide  thirty-five  into  two  such  parts  that  the 
larger  shall  be  nine  more  than  the  smaller.  What  is 
each  part  ? 

24.  George  has  seven  books  more  than  Mary  ;  and 
they  both  have  twejity-nine.     How  many  has  each? 

25.  Two  men,  A  and  B,  were  thirty  miles  apart, 
and  travelled  towards  each  other  till  they  met.  They 
then  found  that  A  had  travelled  six  miles  more  than 
B.     What  distance  did  each  travel  ? 

26.  The  difference  between  two  numbers  is  thir- 
teen, and  their  sum  is  twenty-seven.  What  are  the 
numbers  ? 

27.  There  are  two  numbers,  one  of  which  is  seven 
more  than  the  other,  and  their  sum  is  twenty-three. 
What  are  the  numbers  ? 

28.  The  sum  of  three  numbers  is  thirty-six.     The 


§  4  ]  INTELLECTUAL     ALGEBRA.  29 

first  is  two  more  than  the  second,  and  the  third  four 

more  than  the  second.     What  are  the  numbers  ? 

Let  X  =z  the  second  number ; 

then  the  first  will  be  x  -j-  2, 

and  x-\-4:.  will  represent  the  third. 

Then  x-{-x-\-'2-\-%  -\-  4:  will  express  their  sum. 

But  their  sum  is,  by  the  question,  36  ; 

therefore,  3  x  -|~  ^  ^^  ^^• 

Subtracting  G  from  each  member  of  the  equation,  gives 

3z  +  6  — 6  =  36  — 6; 

uniting  terms,  3  a;  =  30. 

Dividing  each  member  of  the  equation  by  3,  gives 

X  =  10,  the  second  number  ; 

then  a;  -|-  2  =:  12,  the  first  number  ; 

and  x-\-  A=i  14,  the  third  number. 

29.  Three  boys  have  thirty-two  books.  John  has 
three  more  than  James,  and  William  two  more  than 
John.     How  many  has  each  1 

30.  George,  Charles,  and  Robert,  together,  have 
forty-eight  pears.  George  has  "two  more  than  Charles, 
and  Robert  has  as  many  as  George  and  Charles  both. 
How  many  has  each  ? 

31.  The  sum  of  three  numbers  is  fifty-two.  The 
first  is  twice  the  second,  and  the  second  is  four  more 
than  the  third.     What  are  the  numbers  1 

32.  Three  men  spend  sixty-three  dollars.  A  spends 
three  dollars  more  than  C,  and  C  spends  twice  as 
many  as  B.     How  many  dollars  does  each  spend '? 

33.  Divide  the  number  forty-one  into  three  such 
parts  that  the  greatest  shall  be  seven  more  than  the 
least,  and  the  third  four  more  than  the  least.  What 
are  the  parts  ' 


30  INTELLECTUAL     ALGEBRA.  [§  5 


SECTION  V. 

1.  After  giving  away  three  books,  George  founa 
he  had  seven  left.     How  many  had  he  at  first? 

Let  X  represent  his  number  of  books  at  first ; 

then  X  —  3,  that  is,  x  less  3,  will  equal  the  number  he 

had  left. 

But  he  had  7  books  left. 

Therefore,  by  a  condition  of  the  question,  x  —  3  :=  7. 

If  I,  with  3  taken  from  it,  is  equal  to  7,  z  itself  must 

be  equal  to  7  and  3  more ; 

that  is,  z  rrr  7  +  3  ; 

or  a;  :=r  10  books,  the  number  he  had  at  first. 

Or, 

adding  3  to  each  member  of  the  equation. 

z  — 3  — 7, 

gives  z  — 3  +  3  =  7  +  3. 

Uniting  terms  in  each  member,  gives 

a;  z=  10,  as  above. 

2     Anna  lost  four  needles,  and  now  has  nine  left. 

How  many  had  she  at  first  ? 

3.  Charles  gave  seven  cents  to  a  poor  woman,  and 
then  had  twelve  left.     How  many  had  he  at  first  ? 

4.  Robert  lost  twelve  pens,  and  has  seventeen  left. 
How  many  had  he  at  first  ? 

5.  Mary  has  eaten  nine  plums,  and  she  still  has 
sixteen  in  her  basket.     How  many  had  she  at  first? 

6  Thomas  took  seventeen  cents  from  his  purse, 
and  then  there  were  ten  remaining  in  it.  How  many 
cents  were  in  the  purse  at  first  ? 

7.  What  number  is  that,  from  which  if  eleven  be 
taken,  the  remainder  will  be  thirteen  ^ 


§  &•]         INTELLECTUAL  ALGEBRA.  31 

8.  From  what  number  must  seven  be  taken,  that 
thirteen  only  may  remain  '? 

9.  Jane  has  three  less  than  Mary,  and  both  together 
have  eleven  books.     How  many  has  each? 

Let  X  represent  Mary's  share ; 

then  X  less  3,  that  is,  x  —  3,  will  stand  for  Jane's  share, 

and  X  -\-x  —  3,  that  is,  2x^3,  will  represent 

what  both  have. 

But  both  have  11  books; 

therefore,  2  3;  — 3=  11.     ' 

If  2  a;  less  3  is  1 1 ,  2  x  must  equal  1 1  and  3  more, 

that  is,  2  X  =  11  +  3,  which  is  14. 

If  2  X  =  14,  then  x  will  equal  ^  of  14,  which  is  7 ; 

therefore  x  z=  7  books,  or  Mary's  share ; 

and  X  —  3  =:  4  books,  or  Jane's  share. 

Or, 

since  2 X — -Srrill, 

adding  3  to  each  member  of  the  equation,  gives 

2  X  — 34-3=114-3. 

Uniting  terms  in  each  member,  2  x  =  14  ; 

dividing  each  member  by  2,  x  r=  7  ; 

and  X  —  3  =  4,  as  above. 

10.  Lucj  ind  Anna  had  each  an  equal  number  of 
pictures.  After  three  were  taken  from  Anna,  both 
together  had  only  seventeen.     How  many  had  each  ? 

Let  X  represent  Lucy's  number  of  pictures: 
then  X  will  also  equal  the  number  Anna  had   at  first, 
aiid  X  —  3  =  the  number  Anria  had  remaining,  aftei 

three  pictures  wjere  taken  from' her. 

Then  x  -f-  x  —  3  =  the  number  both  together  had,  aftc) 

three  were  taken  from  Anna. 

But  both  had  17,  after  three  were  taken  from  Anna: 


32.  INTELLECTUAL     ALGEBRA.  T^  5^ 

therefore,  by  the  conditions  of"  the  question, 

2x  — 3=  17. 

Adding  3  to  each  member  of  the  equation,  gives 

22;_3-|-3=17-f  3; 

uniting  terms  in  each  member,  2  z  =:  2(>. 

Dividing  eacli  member  by  2,  gives 

x=  10,  or  Lucy's  number  of  pictures, 

uid  X  —  3^7,  or  the  number  Anna  had  remaining. 

11.  Josephine  and  Georgiana  had  each  an  equal 
oidmber  of  clolls  ;  but  Georgiana  gave  away  five  of 
her  dolls,  a;id  now  both  together  have  but  eleven. 
How  many  had  each  at  first,  and  how  many  has  each 
now? 

12.  John  and  William  together  have  twenty  cents. 
John  lacks  only  seven  of  having  twice  as  many  as 
William.     How  many  has  each  ? 

If  a;  =  William's  share,  then  2  x  —  7  =  John's  share. 

13.  Eliza  and  Sarah  bought  a  doll  for  twenty-six 
cents.  If  Sarah  had  paid  four  cents  more,  she  would 
have  paid  twice  as  much  as  Eliza.  How  much  did 
each  pay  ? 

14.  A  man  and  a  boy  together  have  thirty  dollars. 
If  Ihe  man  had  two  dollars  more,  he  would  have  three 
tiiiics  as  many  as  the  boy.  How  many  dollars  has 
each  1 

15.  The  difference  between  two  numbers  is  seven. 
Now,  if  X  be  put  for  the  greater,  what  will  stand  for 
the  smaller  ?     What  will  represent  their  sum  ? 

10.  If  the  sum  of  the  two  numbers  in  the  15tl; 
question  be  thirteen,  what  will  be  the  numbers  ? 

17.  If  the  sum  of  the  two  numbers  in  the  l^fh 
question  be  nine,  what  will  be  the  numbers? 


§  ^'1 


INTELLECTUAL     ALGEBRA.  33 


IS.  The  sum  of  two  numbers  is  twelve,  and  their 
difference  is  four.     What  are  the  numbers  ? 

19.  The  sum  of  two  numbers  is  thirty-three,  and 
the  greater  lacks  three  of  being  equal  to  three  times 
the  smaller.     What  are  the  numbers  ? 

29.  The  sum  of  two  numbers  is  twenty-seven,  and 
the  greater  lacks  twelve  of  being  twice  the  smaller. 
What  are  the  numbers? 

21.  The  sum  of  two  numbers  is  fifty,  and  the 
greater  is  only  ten  less  than  four  times  the  smaller. 
What  are  the  numbers  ? 

22.  The  united  ages  of  a  father  and  son  are  thirty- 
four  years,  and  the  father  lacks  only  two  years  of  being 
five  times  as  old  as  the  son.    What  was  the  age  of  each  ? 

23.  George  and  Mary  together  have  written  forty- 
five  copies  in  school.  If  Mary  had  written  three 
copies  more,  she  would  have  written  three  times  as 
many  as  George  did.     How  many  did  each  write? 

24.  The  sum  of  two  numbers  is  sixty-seven,  and 
the  greater  lacks  five  of  being  seven  times  the  smaller 
What  are  the  numbers  ? 

25.  The  united  ages  of  a  gentleman  and  his  two 
sons  are  fifty-two  years.  The  elder  son  is  three  times 
as  old  as  the  younger  ;  the  father's  age  lacks  but  eight 
years  of  being  twice  the  sum  of  the  united  ages  of 
both  sons.     What  is  the  age  of  each? 

26.  If  X  less  three  is  equal  to  seven,  how  many 
must  be  added  to  seven  to  make  it  equal  to  a;  ? 

27.  If  X-  —  3  be  added  to  x  —  3,  what  will  be  the 
sum? 

3  less  than  x  added  to  3  less  than  x,  will  be  G  less 
than  2  r,  or  2  X  less  (>. 
3 


34  INTELLECTUAL     ALGEBRA.  §     6.' 

28.  If  X  —  3  be   muJtiplied   by  2,  what  will   the 
product  be  1 

29.  If   X  —  3  be   multiplied   by  3,   ,vhat  will   the 
product  be  ? 

30.  If  X  —  3   be   multiplied   by  5,  what  will   the 
product  be  1 

31.  If  2x  —  7  be  multiplied  by  4,  what  will  the 
product  be  ? 

32.  When  x— 8=12,  what  must  be  added  to  12 
that  the  sum  may  be  equal  to  x? 

33.  WJien  x  less  8  plus  8  is  equal  to  12  plus  8, 
what  is  the  value  of  x  ? 


SECTION   VI. 

1.  Adeline  bought  an  equal  number  of  pears  and 
peaches  for  nine  cents,  paying  one  cent  for  each  pear, 
and  two  cents  for  a  peach.  How  many  of  each  did 
she  buy  ? 

Let  X  represent  the  number  of  pears ; 

then  X  will  also  stand  for  the  number  of  peaches. 

She  bought  x  pears,  and  x  peaches. 

She  paid  x  times  1  cent  for  the  pears,  and  x  times  2 

cents  for  the  peaches. 

BuJ.  X  times  1  cent  is  x  cents, 

and  X  times  2  cents,  being  twice  as  many  as  x  times 

1  cent,  will  equal  2  x  cents; 

then  a; -f- '^  ^  ^-^  "^  ^  cents,  will  represent  the  number 

of  cents  she  gave  for  all  the  pears  and  peaches. 


Q  6.]  INTELLECTUAL     ALGEBRA.  35 

But  she  gave  9  cents  for  all  of  them. 
Therefore,  by  the  conditions  of  the  question, 

Dividing  each  member  of  the  equation  by  3,  gives 

a;  =  3,  the  number  of  each  that  she  bought. 

Alls.  3  pears  and  3  peaches. 

2.  George  bought  lead  pencils  at  two  cents,  and 
slate  pencils  at  one  cent  apiece,  of  each  an  equal 
number.  They  cost  him  fifteen  cents.  How  many 
of  each  did  he  buy  1 

3.  A  boy  bought  as  many  pen-holders  as  writing- 
books  ;  paying  three  cents  for  a  pen-holder,  and  six 
cents  for  a  writing-book.  How  many  of  each  did  he 
buy  for  twenty-seven  cents  ? 

4.  A*  farmer  sold  as  many  barrels  of  apples  as  of 
cider,  and  received  for  the  whole  twenty  dollars.  He 
sold  the  apples  at  two  dollars  a  barrel,  and  the  cider 
at  three  dollars  a  barrel.  How  many  barrels  of  each 
did  he  sell  ? 

5.  George  and  Henry  together  paid  twenty-eight 
cents  for  oranges,  and  each  bought  an  equal  number 
But  George  paid  at  the  rate  of  three  cents,  and  Henry 
at  the  rate  of  four  cents,  apiece.  How  many  oranges 
did  each  buy,  and  how  many  cents  did  each  pay  for 
his  oranges  ? 

6.  A  farmer  sold  as  many  sheep  as  calves ;  receiv- 
ing two  dollars  for  a  sheep,  and  four  dollars  for  a  calf 
He  sold  them  all  for  twenty-four  dollars.  How  many 
of  each  did  he  sell  ? 

7.  Two  men  keep  the-  cows  in  a  hired  pasture, 
for  which  they  pay  forty-tw.  dollars  a  year.  One  has 
two  cows   and  the  other  has  ^ve  cows.     How  much 


36  INTELLECTUAL    ALofcBRA.  [§  G. 

does  it  cost  to  Keep   each   cow,  ami    how  many  dollars 
does  each  man  pay  ? 

8.  A  farmer  gave  his  laborers  sixty-three  dollars; 
paying  each  man  six  dollars,  and  each  boy  three  dol- 
lars. There  were  as  many  men  as  boys.  How  many 
boys  were  there? 

9.  Martha  bought  melons  and  oranges,  of  each  an 
equal  number,  for  forty-eight  cents,  giving  ten  cents 
for  a  melon  and  two  for  an  orange.  How  many  of 
each  did  she  buy  ? 

10.  A  man  bought  cows  at  eighteen  dollars  apiece, 
and  sheep  at  two  dollars,  of  each  an  equal  number. 
How  many  of  each  did  he  buy  for  one  hundred 
dollars  ? 

11.  Charles  has  his  money  in  cents  and  half-dimes, 
of  each  an  equal  number.  The  whole  of  his  money 
amounts  to  thirty-six  cents.  How  many  copper  and 
how  many  silver  pieces  of  money  has  he  ? 

12.  Zealous  has  his  money,  amounting  to  fifty-five 
cents,  in  two  kinds  of  coin,  namely,  cents  or  copper 
pieces,  and  dimes  or  silver  piece.s,  worth  ten  of  the 
copper  pieces.  He  has  as  many  silver  pieces  as  he 
has  copper  pieces.  How  many  pieces  of  each  kind 
has  he  ? 

13.  Frederic  has  his  money  in  dimes,  half-dimes, 
and  cents ;  and  he  has  tlie  same  number  of  pieces  oi 
each  kind  of  coin.  The  value  of  all  the  pieces  is 
eighty  cents.     How  many  pieces  of  each  kind  has  he? 

14.  Nichols  bought  an  equal  number  of  apples, 
Jemons,  and  oranges  ;  paying  one  cent  for  an  apple, 
two  for  a  lemon,  and  three  for  an  orange.  How  many 
of  each  did  lie  buy  for  twenty-four  cents  ? 


§  6.1  INTELLECTUAL.      ALGEBRA.  f37 

Let  z  represent  the  number  of  each 

4.8  he  gave  1  cent  for  one  apple,  x  apples  cost  x  cents; 

as  a  lemon  cost  2  cents,  x  lemons  cost  2  x  cents; 

and  X  oranges,  at  3  cents  apiece,  cost  3  x  cents. 

Then  x-\-^x-\-'^x-=zQx  cents,  the  cost  of  all  he 

bought. 

But  he  paid  for  all  24  cents. 

Therefore,  by  the  conditions  of  the  question, 

6  2  =  24  ; 

and,  dividing  each  member  by  6,  x=i^^-, 

that  is,  x  is  ^  of  24,  which  is  4. 

Then  he  bought  4  apples,  4  lemons,  and  4  oranges. 

15.  The  sum.  of  three  numbers  is  sixty-three.  The 
first  is  twice  the  second,  and  the  second  is  twice  the 
third.     What  are  the  numbers  1  j 

16.  A  horse,  saddle,  and  bridle,  together,  cost  one 
hundred  and  forty-four  dollars.  The  saddle  cost  twice 
as  much  as  the  bridle,  and  the  horse  three  times  as 
much  as  the  saddle  and  bridle  together.  What  was 
the  cost  of  each  1 

17.  A  man  deposited  in  a  Savings  Bank,  at  differ- 
ent times,  three  several  sums  of  money,  amounting  to 
seventy  dollars.  The  first  time  he  put  in  twice  as 
much  as  he  did  the  second  time,  and  the  third  time 
twice  as  much  as  he  did  the  first  time.  How  many 
dollars  did  he  deposit  each  time  ? 

18.  A  boy  has  peaches,  pears,  and  apples,  of  each 
an  equal  nurnber  ;  and  they  cost  him  forty-eight  cents. 
He  gave  twice  as  much  for  a  pear,  and  three  times  as 
much  for  a  peach,  as  he  did  for  an  apple ;  and  an 
apple  cost  one  cent.  How  many  had  he  of  each? 
What  did  all  of  each  kind  cost? 


38  INTKLLECTUAI.      ALGEBRA.  [§  6. 

19.  Throe  men,  A,  B,  and  C,  keep  their  cows  in 
the  same  pasture,  and  together  pay  fifty-six  dollars  for 
the  use  of  it.  A  has  one  cow,  B  has  three,  and  C 
has  as  many  as  A  and  B  together.  What  was  the  cost 
for  each  cow,  and  what  did  each  man  pay  ? 

20.  A  boy  has  three  times  as  many  plums  as  pears, 
and  twice  as  many  pears  as  peaches ;  in  all,  fifty-four 
How  many  has  he  of  each  kind  ? 

-i.    Anna,  Lucy,  and  Mary  are  to  share  sixty  dul 
lars.     Anna  is  to  have  two,  and  Lucy  three  dollars,  to 
Mary's  one.     How  many  dollars  will  each  have  ? 

22.  Charles  has  some  apples  ;  George  has  twice  as 
many  as  Charles  ;  and  Peter  three  times  as  many  as 
George.  Now,  if  %  be  put  foj  the  number  which 
Charles  has,  what  will  stand  for  the  respective  shares 
of  George  and  Peter  ?  and  what  will  represent  the 
sum  of  their  shares,  or  the  whole  number  of  apples  ? 

23.  If  the  whole  number  of  apples  in  the  22d 
question  be  ninety,  what  will  be  the  value  of  x,  and 
how  many  apples  has  each  boy  ? 

24.  If  the  value  of  %,  in  the  22d  question,  be  nine, 
how  many  apples  will  each  boy  have  ? 

25.  Robert  gave  one  half  of  his  money  for  quills, 
at  a  cent  apiece,  and  the  other  half  for  quills;  at  the 
rate  of  three  for  a  cent.  He  bought  thirty-six  quills. 
How  much  money  had  he  ? 

Let  x^=.  one  half  of  his  money. 

At  a  cent  apiece,  for  %  cents  he  bought  x  quills. 

At  3  quills  for  a  cent,  for  %  cents  he  got  three'  times 

as  many  quills  as  he  did  at  the  rate  of 

one  quill  for  a  cent. 


§  6.]  INTELLECTTjAl.      ALGEBRA.  39 

Then,  ibr  the  other  half  of  his  money,  he  got  3  x  quills, 

and  he  bought,  in  all,  x  quills  and  3  x  quills. 

But  he  bought  36  quills  ; 

therefore,  x-\-S  x,  or  4x=i  36. 

Dividing  each  member  by  4,  a;  :=r  9  cents,  or  half  at 

his  money. 

Then  2  z  r=  18  cents,  the  whole  of  his  money. 

26.  A  girl  bought  some  oranges  for  forty-five  cents, 
paying  two  cents  apiece  for  one  half  of  them,  and 
three  cents  apiece  for  the  other  half.  How  many  did 
she  buy  at  each  price,  and  how  many  in  all  ? 

27.  A  man  bought  a  number  of  sheep  for  twenty- 
seven  dollars.  Half  of  them  cost  a  dollar  apiece,  and 
the  other  half  two  dollars  apiece.  How  many  did  he 
buy? 

28.  One  man  travelled  four  miles  an  hour,  and 
another  five  miles.  Each  travelled  the  same  number 
of  hours.  The  sum  of  the  distances  which  they  trav- 
elled was  seventy-two  miles.  How  many  hours  and 
miles  did  each  travel  ? 

29.  Two  men  are  seventy  miles  apart,  and  are 
travelling  towards  each  other.  One  travels  three 
miles  an  hour,  and  the  other  four  miles  an  hour.  In 
how  many  hours  will  they  meet?  and  what  distance 
does  each  travel  ? 

30.  A  boy  bought  fifty-four  plums  and  grapes,  pay- 
ing an  equal  sum  of  money  for  each  kind  of  fruit. 
The  plums  cost  at  the  rate  of  a  cent  for  two,  and  the 
grapes  a  cent  for  four.  Hov/  many  of  each  kind  did 
he  buy  ?  and  how  much  money  did  .ill  co-st  1 


40  I-NTEL,L,KC'I  I  AI.      ALGEBRA.  [§  7. 


SECTION    VII. 

1.  Charles  has  half  as  many  b(X)ks  as  Georg< ,  and 
ihey  both  have  nine.     How  manv  books  has  each  ? 
Let  X  =  George's  number  of  books  ; 
then  Charles's  number  will  be  one  half  of  x, 

which  is  X  divided  by  2,  and  it  may  be  written  thus,  —  ; 

then  x-\ will  represent  the  whole  number  of  books 

But  the  whole  number  of  books  is  9  ; 
therefore,  by  the  conditions  of  the  question, 

x-\-~  =  9. 

g  X 

But  X  =i  ^^—,  or  two  halves  of  x, 

2 

,  2  X    ,     X         5x 

and =1  —  : 

2  2  2 

therefore,  ^  r=:  9. 

2 
JC  three  halves  of  a;  are  equal  to  9,  one  half  of  x  must 
be  one  third  of  9,  which  is  3,  or  Charles's  number 
of  books. 
If  one  haJf  of  x  is  3,  the  whole  of  x  will  be  twice  3, 
which  is  6,  or  George's  number  of  books. 
Again, 

3x 
sinc3  —  =  9,  or  one  half  of  3  r  is  9, 

2 

the  whole  of  3  x  is  twice  9,  which  is  18 ; 

therefore,  3  x^  18. 

If  32;==  18,  X  will  be  one  third  of  18,  which  is  6 

then  x=i6,  or  George's  number  of  books, 

and  —  =  3,  or  Charles's,  &.c. 


5   ~  ]  I^fTBHJECTUALl     AJL.GEBRA.  41 

Or, 

3,T  „ 

since  — =^\y, 
2 

iniiltiplying  each  member  of  the  equation  by  2,  gives 

33;  z=  18. 

Dividing  each  member  of  this  last  equation  by  3,  gives 

X  =  6,  as  above. 

2.  If  -^  =  6,  what  will  — ,  or  one  fourth  of  x,  equal  1 

What  will  a-  equal  ? 

3  X 

3.  If  each  member  of  the  equation  — nr  6  be  mul- 
tiplied by  4,  v.hat  equation  will  express  the  product? 

4.  If  each  member  of  the  equation  Sx=z  24,  be 
divided  by  3,  what  equation  will  express  the  q.uotient? 

0.    In  the  equation  —  zn  G,  what  is  the  value  of  a:  ? 

5 

that  is,  what  number  does  x  represent  ? 

6.  In  two  classes  there  are  fifteen  pupils,  and  the 
grammar  class  is  half  as  large  as  the  reading  class. 
How  many  pupils  in  each  class  ? 

7.  Anna  and  Charles  together  have  twenty-seven 
pens,  and  Charles  has  half  as  many  as  Anna.  How 
many  has  each  ? 

8.  Robert  is  half  as  old  as  Mary,  and  the  sum  of 
their  ages  is  twenty-one.     What  is  the  age  of  each  ? 

9.  In  an  orchard  of  thirty  trees  there  are  half  as 
many  cherry-trees  as  pear-trees.  How  many  trees 
are  there  of  each  kind  ? 

10.  The  sum  of  two  numbers  is  eighteen,  and  one 
IS  half  as  large  as  the  other.      What  are  the  numbers  ] 

Let  X  =1  the  larger,  &c. 

11.  Divide   twenty-four  into  two  such   parts,  that 


42  INTELLECTUAL,     ALGEBRA.  rjc  7 

one  sliall  be  hiilf  of  the  other.     What  will  the  parts 
be? 

12.  One  number  is  half  as  large  as  another,  and 
their  sum  is  thirty-nine.     What  are  the  numbers  ? 

13.  What  number  must  be  added  to  half  of  itself, 
that  the  sum  may  be  thirty-three  ? 

14.  W^hat  number  must  be  added  to  half  of  itself, 
that  the  sum  may  be  forty-two  ? 

15.  Add  such  a  number  to  half  of  itself,  that  the 
sum  may  be  thirty.     What  will  the  number  be  ? 

16.  A  number  and  half  of  the  same  number  added 
together  are  thirty-six.     What  is  the  number  ? 

17.  What  number  must  be  added  to  a  third  part 
of  itself,  that  the  sum  may  be  twenty  ? 

18.  The  sum  of  two  numbers  is  thirty-two,  and 
one  is  a  third  part  of  the  other.  What  are  the  num- 
bers ? 

19.  Divide  thirty-five  into  two  such  parts,  that  one 
part  shall  be  one  fourth  of  the  other. 

20.  A  and  B,  in  partnership,  gain  thirty-six  dollars. 
A  put  into  the  firm  half  as  much  money  as  B,  and 
shared  proportionally  in  the  gain.  What  was  each 
one's  share  of  the  gain  ? 

21.  A  man  sold  a  knife  for  thirty  cents,  by  which 
he  gained  one  fourth  of  the  cost.  How  much  did  it 
cost  ? 

22.  In  a  school  of  forty-five  pupils,  there  are  one 
fourth  as  many  girls  as  boys.  How  many  of  each 
sex  ? 

23.  One  number  is  one  fifth  of  another,  and  their 
sum  is  forty-two.     What  are  the  numbers  ? 

24.  A  man  sold  a  lujrse   for  fifty-six  dollars,  by 


§7.] 


INTELLECTCAL     ALGEBRA.  4^1 


which  he  gained  one  sixth  of  what  the  horse  cost 
What  was  the  cost  of  the  horse  1 

25.  A  put  into  the  firm  one  fifth  as  much  monej' 
as  B ;  they  gained  seventy-two  dollars.  What  was 
each  one's  share  of  the  gain  ? 

2G.  What  number  must  be  added  to  one  eighth  of 
itself,  that  the  sijm  may  be  ninety  1 

27.  The  sum  of  two  numbers  is  forty-eight,  and 
one  is  one  seventh  of  the  other.  What  are  the  num- 
bers ? 

28.  If  you  count  the  lambs  with  the  sheep,  you 
will  find  one  hundred  and  eight  in  the  flock,  and 
there  is  one  eleventh  as  many  lambs  as  sheep.  How 
many  of  each  ? 

29.  The  sum  of  two  numbers  is  ninety-nine,  and 
one  is  one  tenth  of  the  other.  What  are  the  num- 
bers ? 

30.  What  number  must  be  added  to  one  ninth  of 
itself,  that  the  sum  may  be  one  hundred  1 

31.  Mary  has  one  sixth  as  many  books  as  Anna, 
and  they  both  have  forty-nine.  How  many  books  has 
each  ? 

32.  Divide  eighty  into  two  such  parts,  that  one 
part  shall  be  one  ninth  of  the  other.  What  will  the 
pafts  be  ? 

33.  Two  men  gained  sixty  dollars,  and  one  gained 
one  fifth  as  much  as  the  other.  What  was  the  gain 
of  each  ? 

34.  A  man,  by  selling  his  cow  for  thirty-two  dol- 
lars, gained  one  seventh  of  what  she  cost  him.  How 
many  dollars  did  she  cost? 

35.  The  sum  of  two  numbers  is  fifty-six,  and  one 


44  INTELLECTUAL     ALGEBRA.  £^  7 

seventh  of  one  number  is  equal  to  the  whole  of  the 
other.     What  are  the  numbers  ? 

Let  X  =:  the  greater  ; 

then  —  =:  the  less. 

7 

■  36.    Two  men  together  have  twenty-five  hundreds 
of  dollars,  and  A  has  one  fourth  as  many  hundreds  as 
B.     How  many  hundreds  has  each  ? 
Let  x=:  the  number  of  hundreds  that  B  has,  &/C. 

37.  A  man  sold  a  house  for  twenty-four  hundreds 
of  dollars,  by  which  bargain  he  gained  one  fifth  of 
what  the  house  cost  him.  How  many  hundreds  of 
dollars  did  he  gain  ? 

38.  There  were  only  ninety-nine  sound  oranges  in 
a  box  bought  by  two  boys ;  but  Daniel  paid  only  one 
eighth  part  as  much  as  Frederic.  How  many  oranges 
ought  each  to  have  1 

39.  What  number  must  be  added  to  one, fifteenth 
of  itself,  that  the  sum  may  be  sixty-four  ? 

40.  A  brother  and  sister  inherit  an  estate  which 
sold  for  sixteen  thousands  -of  dollars ;  but  by  their 
father's  Will,  the  brother  is  to  have  only  one  third  as 
much  as  the  sister.  How  many  thousands  of  dollars 
will  each  have? 

41.  Divide  fifty-six  hundreds  into  two  such  parts, 
that  one  part  shall  be  one  seventh  as  many  hundreds 
as  the  other.     How  many  hundreds  will  each  part  be? 

42.  Anna  and  Mary  are  to  share  twenty  chestnuts, 
and  Mary  is  to  have  two  thirds  as  many  as  Anna. 
How  many  will  each  hare  ? 

43.  John   is  three  fourths  as  old  as  Robert,  and 


^  7.]  INTELLECTUAL    ALGEBRA.  45 

t]ie  sum  of  their   ages  is  twenty- eight.      What   is   tho 
age  of  each  ? 

44.  The  sum  of  two  numbers  is  thirty-two,  and  one 
is  three  fifths  of  the  other.     What  are  the  numbers  ? 

45.  One  number  is  five  sixths  of  another,  and 
their  sum  is  seventy-seven.     What  are  the  numbers'.' 

46.  What  number  must  be  added  to  two  ninths  of 
itself,  that  the  sum  may  be  fifty-five? 

47.  A  man  sold  a  yoke  of  oxen  for  ninety  dollars, 
by  which  he  gained  two  sevenths  of  what  they  cost 
hira.  How  much  did  the  oxen  cost?  and  how  much 
did  he  gain? 

48.  In  a  school  there  are  thirty-nine  pupils,  and 
there  are  five  eighths  as  many  studying  algebra  as 
there  are  studying  arithmetic.  How  many  in  each 
study  ? 

49.  What  number  is  that  to  which  two  thirteenths 
of  itself  must  be  added,  that  the  sum  may  be  sixty  ? 

50.  Divide  fifty-four  into  two  such  parts,  that  one 
part  shall  be  only  two  sevenths  as  large  as  the  other. 
What  will  the  parts  be  ? 

51.  A  horse  and  cow  together  cost  ninety-six  dol- 
lars, and  the  cow  cost  three  fifths  as  much  as  the 
horse.     What  was  the  cost  of  each  ? 

52.  A  and  B,  in  partnership,  gain  eighty-tour  dol- 
lars. A  put  into  the  firm  three  fourths  as  much  money 
as  B,  and  they  cire  to  share  the  profits  in  the  same  pro- 
portion.    How  many  dollars  can  each  have  ? 

53.  The  sum  of  two  numbers  is  forty-five,  and  one 
number  is  four  elevenths  of  *.he  other.  What  are  the 
numbers? 


46  INTELLECTUAL     ALGEBRA. 


[§    7 


54.  What  number  must  be  added  to  five  ninths  of 
itself,  that  the  sum  may  be  twenty-eight? 

55.  If  one  third  of  x  be  added  to  one  third  of  r-. 
how  many  thirds  of  z  will  the  sum  be  ?  and  what 
term  will  express  it  ? 

56.  In  ^  of  a-,  f  of  r,  and  f  of  x,  there  are  how 
many  fourths  of  a;?  and  what  term  will  express  their 
sum  ? 

57.  What  is  the  sum  of  -5^  of  z,  f  of  z,  and  |  of  2;  ? 

58.  In  X  and  f  of  z  there  are  how  many  fifths  of 
r?     What  term  will  express  the  sum? 

59.  If  %  be  added  to  \  of  x,  what  term  will  express 
ihe  sum  ? 

60.  What  term  will  express  the  sum,  if  |^  of  x  be 
added  to  f  of  x  ? 

3  .T        5  X        7  .T 

61 .  If  the  expression,  ^-^  -| 1 — '-,  be  reduced  to 

8  8  8 

one  term,  what  will  express  the  sum? 

It        1 X 

62.  How  many  ninths  of  x  in  2  x  -I h  — "^  ^ind 

•'  '      9      '      9 

what  term  will  express  the  sum  ? 

9  X        7  X 

63.  Reduce  the  expression  z  -| 1-  ■ — ,    that    is 

unite  the  terms  in  one  term.     What  will  that  terra 
be? 

64.  Reduce  to  one  term  the  expression 

-  +  T  +  f  +  T' 

What  will  tlie  term  be  ? 


^  8.]  INTELLECTUAL     ALGEBRA.  47 


SECTION  VIII. 

1.   George  has  four  pears,  which  is  one  half  as 
many  as  Anna  has      How  many  has  Anna? 

Let  X  represent  Anna's  number  of  pears ; 

then  — ,  or  z  divided  by  2,  equals  George's  pears 

But  George  has  4  pears  ; 
therefore,  by  the  conditions  of  the  question, 

2 

If  one  half  of  x  is  equal  to  4,  the  whole  of  x  is  equal 

to  8  ; 

therefore,  Anna  has  8  pears. 

Or, 

smce  —  =  4, 

n>ultiplying  each  member  of  the  equation  by  2,  gives 
a;  :r=  8,  or  Anna's  pears. 

2.  Charles  has  eight  arrows,  which  is  one  third  as 
many  as  George  has.     How  many  has  George? 

3.  A  man  has  ten  oxen,  which  is  two  thirds  of  the 
number  of  his  cows.     How  many  cows  has  he  ? 

4.  One  boy  has  eighteen  oranges,  which  is  three 
fourths  as  many  as  another  boy  has.  How  many  has 
the  other  ? 

5.  Twelve  is  three  fifths  of  what  number  ? 

6.  The  smaller  of  two  numbers  is  twenty,  and  is 
is  five  eighths  of  the  larger.  What  is  the  larger 
number  ? 

7.  A  man  sold  a  cow  for  twenty-tour  dollars,  which 


46  INTi:LI.Ii(;iUAL    ALGEBRA.  [§  8. 

was  six   sevenths  of  what   sho  co-t   !iim.     How  much 
«lid  she  cost  him,  and  what  did  he  lose  ? 

8.  In  a  pasture  there  are  six  horses,  which  is  two 
ninths  of  the  number  of  sheep  in  the  same  pasture. 
How  many  sheep  are  in  the  pasture  ? 

9.  George  has  sixty-four  cents,  which  is  four  fifths 
of  the  money  in  Edward's  purse.  How  many  cents 
are  in  Edward's  purse? 

10.  One  number  is  five  eighths  of  another,  and  the 
smaller  is  thirty-five.     What  is  the  larger  number  ? 

11.  There  are  two  numbers,  the  smaller  being 
seven  twelfths  of  the  other,  and  the  smaller  is  twenty- 
one.     What  is  the  larger  number  ? 

12.  A  man  sold  his  horse  for  seventy-two  dollars, 
which  was  eight  nintTis  ef  what  the  horse  cost  him. 
How  «iuch  did  the  lior.se  cost  ?  and  how  much  did  he 
lose  by  his  bargain  ? 

13.  A  part  of  William's  money  is  twenty-four 
cents,  and  he  says  it  is  three  sevenths  of  all  he  has. 
How  much  has  lie  1 

14.  Tlie  smaller  of  two  numbers  is  twenty-eight, 
and  it  is  seven  twelfths  of  the  larger.  What  is  the 
larger  ? 

15.  Charles  has  thirty  cents  in  his  hand,  and  this 
is  five  sixths  of  the  sum  in  his  purse.  How  many 
cents  are  there  in  his  purse  ? 

•16.  One  number  is  four,  and  it  is  two  thirteenths 
of  another.     What  is  the  other  number  ? 

17.  Anna  gave  away  six  books,  which  w-as  two 
ninths  of  all  she  had.     How  many  had  she  ? 

18.  George  gave  to  a  poor  woman  fourteen  cents, 


§8.. 


INTELLECTUAL,      ALGEBRA.  49 


which  was  seven  eighths  of  all  he  had.     How  many 
had  he  ? 

19.  IIow  many  fifths  of  o-  are  there  in  the  whole 
of  xl 

20.  If  two  fifths  of  X  be  taken  from  the  whole  of  .c, 
how  many  fifths  of  :f  will  remain  ? 

21.  A  boy,  after  eating  two  fifths  of  his  plums, 
found  he  liad  twelve  plums  remaining.  What  num- 
ber had  he  at  first,  and  how  many  did  he  eat? 

22.  In  the  whole  of  x  there  are  how  many  elev- 
enths of  x  ? 

23.  If  from  eleven  elevenths  of  x,  eight  elevenths 
of  -a:  be  taken,  vvhat  part  of  x  will  remain  ? 

24.  A  farmer  took  a  load  of  melons  to  market, 
and,  after  selling  eight  elevenths  of  them,  found  he 
had  twenty-one  remaining.  How  many  did  he  take 
to  market  1 

25.  Six  feet  of  the  length  of-  a  pole  are  under 
ground,  and  two  thirds  of  the  pole  are  above  ground. 
How  long  is  the  pole  1 

26.  Twenty-eight,  the  smaller  of  two  numbers,  is 
four  ninths  of  the  other.     What  is  the  larger  number  1 

27.  A  pole  is  three  fourths  under  water,  and  there 
are  seven  feet  out  of  water.     How  long  is  the  pole  ? 

28.  A  man  sold  a  cow  for  three  fourths  of  what 
she  cost  him,  and  by  so  doing  lost  six  dollars.  What 
did  she  cost,  and  for  how  much  did  he  sell  her  ? 

29.  A  man,  going  a  journey,  travelled  three  fifths 
of  the  distance  before  dinner,  and  by  going  twenty 
miles  after  dinner,  he  finished  his  journey.  How 
long  was  the  journey  ? 

30.  A  man  sold  a  horse  for  nine  tenths  of  whnf  <K'! 


50  INTELLKCTLAL      ALfiEBRA.  [&  g 

horse  cost  him,  and  by  the  bargain  lie  lost  ten  dollars. 
What  did  the  hurse  cost ''  ;nid  for  how  much  was  he 
sold  ? 

31.  Eiglit  is  one  third  of  what  number  ? 

32.  Nine  is  three  fourths  of  what  number  ? 

33.  A  boy  gave  away  four  fifths  of  his  money,  and 
nad  six  cents  remaining.  How  much  monty  had  he 
at  first  ? 

34.  William  is  nine  years  old,  and  his  age  is  three 
fifths  of  Mary's.     How  old  is  Mary  ? 

35.  A  boy,  by  permission  of  his  father,  sold  his 
sled  for  nine  elevenths  of  what  it  cost  him,  and  by 
doing  so  he  lost  twenty-four  cents.  What  did  the 
sled  cost  ?   and  for  how  much  did  he  sell  it  ? 

36.  A  rnan,  after  spending  five  eighths  of  his 
money,  found  he  had  twenty-one  dollars  remaining. 
How  much  money  had  he*  at  first  1 

37.  A  girl  took  four  ninths  of  her  money  from  her 
purse,  and  then  there  were  thirty  cents  remaining  in 
it.     How  much  money  was  in  the  purse  at  first  ? 

38.  A  teacher,  having  dismissed  ten  of  his  pupils, 
found  only  three  fifths  of  his  school  remaining.  Of 
how  many  pupils  did  the  school  consist  ? 

39.  A  man  sold  a  carriage  for  four  fifths  of  what  it 
cost  him,  and  by  so  doing  he  lost  one  hundred  dol- 
lars. What  did  it  cost  him  ?  and  for  how  much  did 
he  sell  it  ? 

40.  One  number  is  four  sevenths  of  another  num- 
ber, and  the  smaller  is  sixteen.  What  is  the  larger 
number  ? 

41.  Sixteen  is  four  elevenths  of  what  number? 

42.  Twenty-seven  is  nine  tenths  of  what  uumber? 


^  8.]  INTKLl  KCTUAL     ALGEBRA.  51 

43.  One  mnnber  is  four  ninths  of  another,  and 
their  difference  is  fifteen.  What  is  the  larger  nam 
ber  ? 

Let  x=:the  larger  number, 

4  a 

and  —  will  be  the  smaller  number. 
9 

Then  — ^  :=  the  difference  between  the  numbers. 

9 

But  the  difference  is  15  ; 
therefore,  by  the  conditions  of  the  question, 

9 
Dividing  each  member  by  5,  gives 

—  =  3. 

9 

Multiplying  each  member  by  9,  gives 

Then  the  larger  number  is  27, 

and  —  =  12,  the  smaller  number. 
9  ' 

44.  If  you  saw  off  from  a  sloop's  mast  two  ninths 
of  its  whole  length,  and  find  the  remainder  seventy 
feet  long,  how  long  was  the  mast  at  first  ? 

45.  If  you  take  from  a  basket  of  apples  two 
sevenths  of  the  whole  number  in  it,  fifty  apples  will 
remain  in  it.     How  many  apples  were  in  it  at  first? 

46.  One  number  is  three  tenths  of  another,  and 
their  difference  is  forty-nine.     What  are  the  numbers? 

47.  Henry  sold  three  eighths  of  his  hens,  and  had 
fifteen  remaining.  IIoW  many  had  he  before  he  sold 
any? 

^8.    Forty-eight  is  six  elevenths  of  what  number  ? 
49.    Sixty-three  is  seven  ninths  of  what  number  ! 


b'2  INTELLKCTL'AL     ALGEDRA.  [§  8. 

50  One  man  is  three  fourths  as  old  as  another 
and  the  difference  of  their  ages  is  tea  years.  What 
is  the  age  of  each  ? 

Let  a;  =  the  age  of  the  ehler,  6lc. 

Then  -^;r=  the  difference  of  their  ages. 

51.  One  drove  of  cattle  is  three  fifths  of  another, 
and  the  difference  between  the  two  droves  is  twenty- 
four.     How  many  are  there  in  each  drove  ? 

52.  In  one  flock  there  are  seven  eighths  as  many 
sheep  as  there  are  in  the  other  flock,  and  the  smaller 
flock  lias  in  it  ten  less  than  the  larger.  How  many 
m  each  flock  ? 

53.  Charles  lost  three  eighths  of  his  money,  and 
then  had  only  forty-five  cents.  How  much  money 
had  he  before  his  loss  ? 

54.  One  number  is  three  fourths  of  another,  and 
the  difference  between  them  is  six.  What  are  the 
numbers  ? 

55.  One  number,  is  four  fifths  of  another,  and  if 
seven  be  added  to  the  smaller.  It  will  be  equal  to  the 
larger.     What  are  the  numbers  ? 

50.  The  difference  between  two  numbers  is  twenty- 
four,  and  one  is  three  elevenths  of  the  other.  What 
are  the  numbers  ? 

57.  To  three  fourths  of  what  number  must  five  be 
added   that  the  sum  may  be  equal  to  that  number  ? 

58.  If  to  one  third  of  some  number  twelve  be 
added,  the  sum  will  be  equal  to  the  number  itself. 
What  is  the  number  ? 

59.  If  five  be  added  to  six  sevenths  of  a  man's 
age,  the  sum  will  be  his  age.     How  old  is  he  1 


[§    8. 


INTELLECTUAL     ALGEBRA.  03 


60.  Anna's  age  is  three  fifths  of  Mary's,  and  the 
difference  of  their  ages  is  four.     What  are  their  ages? 

61.  The  difference  between  two  numbers  is 
eighteen,  and  one  is  five  eighths  ef  the  other.  What 
are  the  numbers  ? 

62.  Thirty  is  five  sevenths  of  wh.at  number  1 

63.  Sixty-six  is  eleven  twelfths  of  what  number  ? 

64.  What  part  of  x  must  be  added  to  two  thirds 
of  X,  that  the  sum  may  equal  the  whole  of  a;  ? 

65.  What  part  of  x'  must  be  added  to  three  sevenths 
of  r,  that  the  sum  may  be  the  whole  of  a;  ? 

66.  In  one  half  of  x  there  are  how  many  fourths 
of  xl 

67.  In  one  third  of  x  there  are  how  many  sixths 
of  X? 

68.  How  many  sixths  of  x  are  there  in  two  thirds 
of  x? 

69.  How  many  ninths  of  x  in  one  third  of  a;  ? 

2  X 

70.  Two  thirds  of  x  may  be  expressed  thus,  — ; 

that  is,  2  X  divided  by  3.     How  will  you  express  three 

ninths  of  x  ? 

2  r 

71 .  How  many  ninths  of  x  in  two  thirds  of  x,  or  —  f 

4  .r 

72.  In  —  there  are  how  many  tenths  of  x  ? 

5 

73.  How  many  twelfths  of  x  are  there  in  ^-? 

2  X 

74.  In  -^  there  are  how  many  fourteenths  of  x  ? 

75.  How  many  fourteenths  of  x  are  there  in  —  ? 

^  2 

^  X 

76.  In  ^^-  there  are  how  many  eighteenths  of  x? 


»4  INTELLECTUAL.     ALGEBRA.  [§  9. 


SECTION   IX. 

I.'Anna  gave  away  half  of  her  books,  and  then 
had  four  remaining.     How  many   books  had  she   at 
first? 
Let  X  represent  the  number  of  books  she  had  at  first , 

then    -^,  or   one   half  of  x,  will  represent  what  she 
gave  away ; 

and   X ,  or  a;  less  one  half  of  x,  will  express  the 

number  of  books  she  had  remaining. 

But  she  had  4  books  left ; 

therefore,  by  the  conditions  of  the  question, 

2:  less  one  half  of  a;  =^4. 

Since  x  is  equal  to  two  halves  of  x,  the  equation  is, 

two  halves  of  x  less  one  half  of  a;  rr:  4  ; 

then  one  half  of  a;  =:  4  ; 

consequently,  the  whole  of  x  must  be  twice  4 ; 

and  a;  =  8,  the  books  Anna  had. 

Or, 

X  . 

smce  x =  4, 

multiplying  each  member  of  the  equation  by  2,  give.i 
2x  — a;  =  8; 
uniting  the  terms  in  the  last, 
a;  nr  8,  as  above 
2,    if  from  some  number  one  ha-f  of  itself  be  sub- 
tracted, the  remainder  will  be  eight.      What  is  the 
number  ? 

3    The  difference  between  the  whole  of  a  number 


§9.] 


INTELLECTUAL     ALGEBRA.  55 


and  one  half  of  the  sam«  number  is  six.     Wliat  is  tho 
number  ? 

4.  Charles  lost  one  third  of  his  money,  and  had  six 
cents  remaining.     How  many  cents  had  he  at  first  ? 

5.  If  from  some  number  one  third  of  itself  be  sub 
tracted,  the  remainder  will  be  ten.  What  is  the 
number  ? 

G.  After  Charles  had  spent  one  fourth  of  his  money 
for  a  lead  pencil,  he  had  twelve  cents  remaining  in 
his  purse.  How  much  money  did  the  purse  contain 
before  his  purchase  ? 

7.  If  from  some  number  one  fourth  of  itself  be 
subtracted,  the  remainder  will  be  fifteen.  What  is 
the  number  ? 

8.  A  man  sold  one  sixth  of  his  cows,  and  then  had 
ten  cows  remaining.     How  many  had  he  at  first? 

9.  A  boy  eat  two  thirds  of  his  oranges,  and  had 
four  remaining.     How  many  had  he  at  first? 

Let  a;:=:  his  number  of  oranges; 

then  X  less  two  thirds  of  x,  will  express  the  numbei 

remaining. 

But  he  had  4  oranges  remaining  ; 

therefore,  by  the  conditions  of  the  question, 

2a;  . 

X =z  4. 

3 

Multiplying  each  member  of  this  equation  by  3,  gives 

3a:  — 2a;=12; 

uniting  terms,  x  =  12,  his  number  of  oranges. 

10.  If  two  fifths  of  some  number  be  subtracted 
from  the  whole  of  the  same  number,  the  remainder 
will  be  nine.     What  is  the  number  ? 

11.  The  difference  between  the  whole  of  a  number 


56  INTEJ.LECTUAI,     ALCTEBRA.  [^  9 

and  three  fifths  of  the  same  »uuibcr  is  eight.     What 
is  the  number  ? 

12.  A  mail  sold  a  cow  for  fifteen  dollars,  and  b}-  so 
doing  he  lost  two  fifths  of  what  the  cow  codt.  iJLow 
much  did  the  cow  cost?   and  what  did  he  lose? 

13.  What  number  is  that,  from  which  if  three 
sevenths  of  itself  be  subtracted,  the  remainder  will 
be  twelve? 

14.  A  boy  spent  five  ninths  of  his  money,  and  then 
had  sixteen  cents  remaining.  How  much  money  had 
he  at  first  ? 

15.  If  seven  tenths  of  some  number  be  subtracted 
from  the  number  itself,  the  remainder  will  be  fifteen. 
'Vhat  is  the  number  ? 

16.  A  man  lost  two  ninths  of  the  cost  of  his  horse, 
by  selling  him  for  fifty-six  dollars.  What  did  the 
horse  cost  him  1   and  how  much  did  he  lose  1 

17.  The  difference  between  the  whole  and  five 
eighths  of  a  number  is  eighteen.    What  is  the  number  ? 

18.  A  farmer  sold  three  eighths  of  his  flock  of 
sheep,  and  had  forty  sheep  remaining.  How  many 
sheep  v/ere  in  the  flock  at  first  ? 

19.  If  from  a  number  three  fourths  of  itself  be 
subtracted,  the  remainder  will  be  seven.  W^hat  is  the 
number? 

20.  If  from  Caroline's  age  you  take  three  fourths 
of  her  age,  the  remainder  will  be  six  years.  How 
old  is  Caroline  ? 

21.  From  what  number  must  you  take  three  fifths 
of  itself,  that  the  remainder  may  be  twenty  ? 

22.  The  difference  between  Samuel's  age  and  two 
fifths  of  his  age  is  fifleen.     What  is  his  age  ?> 


S  9.]  INTELLECTUAL     ALGEBRA.  5'. 

23.  The  difference  between  a  number  and  fivt- 
ninths  of  itself  is  sixteen.     What  is  the  number  ? 

24.  A  man  paid  away  three  eighths  of  his  money, 
and  had  thirty  dollars  remaining.  How  much  money 
had  he  at  first? 

25.  The  difference  between  the  whole  of  a  num- 
ber and  seven  tenths  of  it  is  eighteen.  What  is  the 
number  1 

26.  A  boy  spent' four  ninths  of  his  money,  and  had 
thirty  cents  remaining,     llow  much  had  he  at  first  ? 

27.  After  paying  a  debt  with  three  tenths  of  his 
money,  a  man  found  he  had  forty-two  dollars  remain- 
ing. How  much  money  had  he  at  first  ?  and  what 
was  the  debt  ? 

28.  From  what  number  must  four  sevenths  of  it- 
«  self  be  taken,  that  the  remainder  may  be  eighteen? 

29.  The  difference  between  a  number  and  one 
sixth  of  itself  is  ten.     What  is  the  number  ? 

30.  If  one  third  of  x  be  taken  from  the  whole  oi 
z,  how  many  thirds  of  x  will  remain? 

X  :=  — ^,  that  is,  three  thirds  of  x,  or  3  x  divided  by  3 

And   3  X  divided    by   3,   less    x  divided    by   3  r=  2 :. 
divided  by  3 ; 

or,  uniting  terms,  — =^  '^-. 

Ans.  f  of  X. 

31.  What  term  will  express  the  difference  between 

2  X 
X  and  two  thirds  of  x ;  that  is,  between  x  and  —  ? 

3 

32.  If  one  fourth  of  x  be  taken  from  x,  what  will 
txpress  the  remainder? 


58 


INTELLECTUAL     ALGEBRA.  [^  9. 


3  X 

33.  If  —  be  taken  from  x,  what  will  be  the  re 
mainder  ? 

34.  What  is  the  difference  between  x  and  two  fifths 
of  X? 

4  X 

35.  If  —  be  taken  from  x,  what  will  express  the 

remainder  ? 

7  r 

36.  If  — ^  be  taken  from  2  x,  what  term  will  express 

5 

the  remainder? 

c.  10  X  ,10  a;        1 X        3x,  ., 

vx=z  —  :   and =  — ,  the  remamder. 

5    '  5  5  5  ' 

37.  What  will  express  the  difference  between  2  x 
and  —  ? 

7 

38.  What  will  represent  the  difference  between  2  z 
and—? 

7 

7  a- 

39.  If  —  be  taken  from  3  x,  what  will  represent 

the  remainder  ? 

8  X 

40.  In  the  equation'^ =  4,  what  does  x  rep- 
resent ? 

41. 
value  of  X  ;  that  is,  what  number  does  x  represent  ? 

42.  In   tl 
value  of  X  ? 

43.  In  th 
the  value  of  i? 


7  X 
41.     In    the   equation   3z =  6.  what  is  the 


6  X 

42.    In   the   equation   4  i ^  z=  9,  what   is   the 


4^8 

43.   In  the  equation  2  x =  —  or  2f ,  what  is 


(!j   10.]  INTELLECTUAL     ALGEBRA.  59 


SECTION   X. 

1.  George  gave  one  half  of  his  bookri  i,,  iiis  sister, 
and  one  fourth  of  them  to  his  brother,  and  the  i  found 
he  had  given  away  six  books.  How  many  had  George 
at  first  ?  and  how  many  did  he  give  to  each  1 

Let  z  =  the  number  of  books  George  had  at  first ; 
then  — ,  or  one  half   of  x,  expresses   the  number  of 

books  which  he  gave  to  his  sister, 
and  — ,  or  one  fourth  of  x,  expresses  the  number  of 

books  which  he  gave  to  his  brother. 

But  he  gave  them  6  books  ; 

therefore,  by  the  conditions  of  the  question, 

one  half  of  x  and  one  fourth  of  x  =  6  ; 

But  one  half  of  x  is  equal  to  two  fourths  of  x, 

and  two  fourths  of  x  added  to  one  fourth  of  a;  =  three 

fourths  of  x ; 

therefore,  three  fourths  of  a;  zr  6. 

Since  th'ee  times  one  fourth  of  x  is  6,  once  one  fourth 

of  X  is  one  third  of  six,  which  is  two. 

If  one  fourth  of  x  is  two,  the  whole  of  x  is  four  times 

two,  which  is  eight ; 

then  X  :=  8,  the  books  George  had. 

He  gave  one  half  of  x  books,  which  is  four,  to  his  sister. 

He  gave  one  fourth  of  x  books,  which  is  two,  to  hia 

brother. 

Or, 

by  the  conditions  of  the  question, 

4  +  -  =  6. 


^30  INTtLLrXTUAL     ALGEBRA.  [§   10 

Multiplying  each  terra  in  each  member  of  the  equa- 
tion by  4,  gives 

-  +  ^  =  24. 

Reducing  fractions  and  uniting  terms,  gives 
3  z  =  24. 

Dividing  each  member  of  this  last  equation  bj  3,  gives 

—  =  — ,  or  I  ^  fc,    George  s  books  ; 

—  =4,  the  number  he  save  to  his  sister  ; 
2 

—  =z  "2,  ttie  number  he  gave  to  his  brother. 

2.  If  one  half  of  a  number  be  added  to  one  fourth 
of  the  same  number,  the  sum  will  be  nine.  What  is 
the  number  ? 

3.  George  expended  two  thirds  of  all  his  money 
for  a  writing-book,  and  one  sixth  of  it  for  a  pencil. 
He  paid  ten  cents  for  both.  How  much  money  had 
he  at  first? 

4.  If  half  of  a  number  be  added  to  one  fourth  of 
the  same  number,  the  sum  will  be  twelve.  "What  is 
the  number  ? 

5.  If  half  of  a  number  be  added  to  the  same 
number,  the  sum  will  be  fifteen.  What  is  the  num- 
ber ? 

6.  If  two  thirds  of  a  number  be  added  to  one  sixth 
of  the  same  number,  the  sum  will  be  fifteen.  AVhat 
is  the  number  ? 

7.  Charles  eat  one  fourth  of  his  chestnuts,  and 
gave  away  one  third  of  them.  He  then  found  he  had 
fourteen  less  than  at  first.     How  many  had  he  ? 


^   10.]  INTELLECTUAL     ALGEBRA.  61 

Let  %  represent  the  number  he  had  ; 
then,  by  the  conditions  of  the  question, 

-  +  -=14. 

„         X         3  J  1     X         43^ 

IJUt  —  rrr — ,    and    — z= - 
4  12'  3  1.: 

therefore, 1 =  14.' 

12     '     12 

7  X 
Unitinor  terms,  —  ^=- 14. 

12 

Multiplying  each  member  of  the  equation  by  12,  gives 

7i=168. 

Dividing  each  member  of  the  last  equation  by  7,  gives 

rr=24,  the  number  Charles  had. 

Or, 

since 1 =  14  ; 

4*3 

multiplying  each  terra  of  each  member  by  12,  gives 
lif  .1^^108. 

4      '      3  . 

Reducing  fractions,  gives 

3  a; +  4  a;  =168. 

Uniting  terms,  gives 

7x=:16S. 

Dividing  each  member  by  7,  gives 

1  =  24,  as  above. 

8.  If  one  fourth  of  a  number  be  added  to  one  third 
of  the  same  number,  the  sum  will  be  twenty-one. 
What  is  the  number  ? 

9.  The  sum  of  one  fourth  and  seven  twelfths  of  the 
same  number  is  twenty.     What  is  the  number  \ 

10.  If  two  thirds  of  a  number  be  added  to  hilf  of 


H 


62  INTELLECTUAL     ALGEBRA.  [§  10. 

the  same  number,  the  sum  will  be  fourteen.     What  ia 
tlie  number  ? 

11.  A  boy  expended  one  fourth  of  his  money  for 
pens,  and  three  eigliths  of  it  for  pencils.  He  spent 
fifteen  cents.  How  many  cents  had  he  at  first  ?  and 
how  many  had  he  left  ? 

12.  If  two  fifths  of  some  number  be  added  to  three 
tenths  of  the  same  number,  the  sum  will  be  forty-two. 
What  is  the  number  1 

13.  Mary  gave  one  third  of  her  books  to  Jane,  and 
one  seventh  of  them  to  Lucy.  She  gave  away  twenty. 
How  many  had  she  at  first  1 

14.  If  one  third,  one  fourth,  and  one  sixth  of  a 
number  be  added  together,  the  sum  will  be  fifty-four. 
What  is  the  number  1 

15.  Robert  gave  one  fourth  of  his  money  for  pen- 
cils, two  fifths  of  it  for  writing-books,  and  one  tenth 
of  it  for  pens.  He  paid  out  thirty  cents.  How  much 
money  had  he  at  first  ?  How  much  did  he  pay  for 
each  ? 

16.  If  you  add  together  ^,  f ,  and  |  of  some  num 
her,  the  sum  will  be  forty-six.     What  is  the  number  ? 

17.  If  you  add  together  f,  ^,  and  -^^  of  some 
number,  the  sum  will  be  sixty.  What  is  the  num- 
ber? 

18.  The  sum  of  ^,  ^,  and  ^  of  a  school  is  twenty- 
two.     How  many  pupils  are  there  in  the  school  ? 

19.  What  is  the  sum  of  --f -  +  — ? 

20.  What  will  express  in  one  term  the  sum  of  the 
three  terms 1 1 1 

3     '      G      '     12 


^   10.1  INTELLECTUAL     ALGEBRA.  63 

21.    Reduce  the  following  expression  lo  one  term, 
5  "■     2  "'     lO' 


-  -4-  —  -1 — -.     What  will  represent  the  sum  ? 


5  ,x         2  X  X 

22.  Reduce  the  expression  —  -f-  -^r  +  — ,  to  one 
term.     What  will  be  the  sum  ? 

3  X  X 

23.  If  each  member  of  the  equation 1 =  13 

be   multiplied   by  3,  what   equation  will   express   the 
product  1 

9  X 

24.  If  each  member  of  the  equation \-x=z  39, 

be   multiplied   by  4,  what  equation  will   express   the 
product  ? 

25.  To  what  equation  will  9  x-\-  4:X=  156  be 
equal,  when  the  two  terms  of  the  first  member  are 
united  ? 

26.  If  each  member  of  the  equation  13  x  =  156, 
be  divided  by  13,  what  number  will  express  the  value 

of  X? 

X         2  a; 

27.  Reduce    the    equation 1 =7.       What 

number  will  express  the  value  of  a;  ? 

28.  Reduce   the    equation   ~ -\- ~  =z  13.     What 

number  does  z  represent  ? 

X         2  X  X 

29.  Reduce    the     equation 1 1 :=  17 

1  2     '      3      '      4 

What  will  be  the  number  represented  by  a;  ? 

3  X         X 

30.  In   the    c(iuation 1 — ^r=ll,   what   is   the 

4  G 

value  of  a;  ? 


G4  INTKLLECTUAL     ALGEBRA.  [§   I  ' 


SECTION   XI 

1.  Geouge,  Anna,  and  Charles  liave  sixteen  books. 
Anna  has  two  more  than  Charles,  and  George  has  as 
many  as  botli.     How  many  has  each  ? 

Let  X  =  the  number  Charles  has ; 
then  X  -|-  2  =  the  number  Anna  has,  because  she  has 

2  more  than  Charles, 
and  X  added  to  x  -j-  2  :=  the  number  George  has,  be- 
cause he  has  as  many  as  both. 
Then  2  x  -J-  2  =  George's  number  ; 
and  X  added  to  x-\-^  added  f o  2  x  -(- 2,  will   express 
what  they  all  have. 
But  they  all  have  16  books ; 
therefore,  by  the  conditions  of  the  question 
x  +  x  +  2  +  2x-}-2  =  16. 
Uniting  terms,  gives 
4x  +  4  =  16. 
Subtracting  4  from  each  member  of  the  equation,  gives 
4x=12. 
Dividing  each  member  of  the  equation  by  4,  gives 
X  rr:  3,  the  number  Charles  has  ; 
then  X  -|-  2  =  5,  the  number  Anna  has  ; 
and  2x-j-2  =  8,  the  number  of  books  George  has. 

2.  Three  men  have  twenty-two  dollars.  The  first 
has  three  dollars  more  than  the  second,  and  the  third 
has  as  many  as  both  of  the  others.  How  many  dol- 
lars has  each  ? 

3.  The  sum  of  three  numbers  is  forty-four.  The 
first  is  four  more  than  the  second,  and  the  third  is  as 
large  as  both  the  others.     What  are  the  numbers  ? 


^  11]        INTELLECTUAL  ALGEBRA.  05 

4.  Mar},  Lucy,  and  Jane  have  fifty-two  quills. 
Jane  has  five  more  than  Lucy,  and  Mary  has  two 
inoce  than  both.     How  many  quills  has  each  ? 

5.  Jbhn  is  six  years  older  than  James,  and  Wil- 
liam's age  is  three  years  more  than  the  united  ages  of 
both.  The  sum  of  their  ages  is  fifty-one.  What  is 
the  age  of  each  1 

6.  One  number  is  eight  more  than  a  smaller  num- 
ber, and  a  third  number  is  four  more  than  both,  and 
the  sum  of  the  three  numbers  is  sixty-eight.  What 
are  the  numbers  ? 

7.  George  is  nine  years  older  than  William,  and 
Herman  is  two  years  older  than  both.  The  sum  of 
their  ages  is  forty.     What  is  the  age  of  each  1 

8.  Three  men  received  seventy-seven  dollars.  A 
received  three  dollars  more  than  B,  and  B  received 
four  dollars  more  than  C.  LIow  many  dollars  did 
each  receive  ? 

9.  The  sum  of  three  numbers  is  forty.  The  first  is 
ten  more  than  the  second,  and  the  third  is  seven  less 
than  the  first.     What  are  the  numbers? 

Let  X  =■  the  second  number  ; 

then  X  -\-  10  z=z  the  first  number. 

The  sum  of  the  two  is  2  a;  -|-  10. 

The  third  number,  being  7  less  than  the  first,  will  be 

expressed    by   a; -|- 10  —  7,    or    x-|-3 

But  the  sum  of  the  three  is  40  ; 

therefore,  x-]-x-^10-\-x-{-3=z40. 

Uniting  the  terms  of  the  first  member,  gives 

3x  +  13rz:40. 

Taking  13  from  each  member,  gives 

3  2:  =  27. 

5 


G6  INTELLECTUAL     ALGEBRA.  [§11 

Dividing  each  member  by  3,  gives 

X  =r  9,  the  secaiMi  number  ; 

a;  -f"  10  ^^  19)  the  first  number  ; 

x  -|-  3  =  12,  the  third  number. 

10.  Three  men  added  their  ages,  and  found  the 
sum  was  one  hundred  years.  The  first  was  twelve 
years  okler  than  the  second,  and  the  age  of  the  third 
was  eight  years  less  than  the  sura  of  the  ages  of  the 
first  and  second.     How  old  was  each  1 

11.  One  number  is  seven  more  than  a  second  nuni 
her,  and  a  third  is  nine  less  than  the  sum  of  the  other 
two.  Their  sum  is  fifty-three.  What  are  the  num- 
bers ? 

12.  There  are  ninety  sheep  in  a  flock,  owned  by 
three  men.  A  owns  twelve  more  than  B,  and  C  owns 
fourteen  less  than  A  and  B  both.  How  many  sheep 
does  each  own  ? 

13.  The  sum  of  three  numbers  is  seventy.  The 
first  is  eleven  more  than  the  third,  and  the  second  i& 
as  much  as  the  first  and  third  together,  lacking 
twelve     What  are  the  numbers  ? 

14.  Three  men  together  have  eighty-two  dollars. 
A  has  fifteen  more  than  B,  and  C  has  as  many  as  A 
and  B  both.     How  many  dollars  has  each? 

15.  Henry,  William,  -and  Robert,  together,  have 
forty-two  cents.  Henry  has  eight  more  than  W^illiam, 
and  Robert  has  only  six  less  than  both.  How  many 
cents  has  each  ? 

16.  In  the  equation  r-\-2x-\-G  —  8=  16,  what  is 
the  value  of  a;  ? 

17.  What  is  the  value  of  x  in  the  equation  x-|-3 
-J-2a--f8-|-z  =  47? 


§   li    ]  INTELLECTUAL     ALGE,BRA.  G7 

IS."  What  is  the  value  of  x  in  the  equation  x-\-'\lx 
—  13  +  a;  +  4  =  ll? 

19.  Fifteen  books  are  to  be  divided  into  three 
such  parts,  that  John  shall  have  four  more,  and  Wil- 
liam four  less,  than  Robert.  How  many  will  each 
have  ? 

20.  Divide  thirty-nine  into  three  such  parts,  that 
the  first  shall  be  seven  greater,  and  the  second  seven 
less,  than  the  third.     What  are  the  parts  ? 

21.  Lucy,  Mary,  and  Anna,  received  each  the  same 
number  of  oranges,  in  all  eighteen.  But  Anna  gave 
three  of  her  oranges  to  Lucy.  Then  how  many  had 
each? 

22.  Divide  twenty-three  into  three  such  numbers, 
that  the  first  shall  be  five  more,  and  the  third  three 
less,  than  the  second.     What  are  the  numbers  ? 

23.  Three  men  are  to  receive  thirty-eight  dollars. 
The  first  is  to  have  seven  more,  and  the  second  five 
less,  than  the  third.  How  many  dollars  will  each 
receive  ? 

24.  The  sum  of  three  numbers  is  twenty-five.  The 
first  is  four  more,  and  the  second  six  less,  than  the 
third.     What  are  the  numbers  ? 

25.  A,  B,  and  C,  gained  twenty-seven  dollars,  and 
they  are  to  share  in  the  following  proportions;  A  is 
to  have  four  less  than  B,  and  C  five  less  than  B.  How 
many  dollars  will  each  receive  ? 

26.  The  sum  of  three  numbers  is  twenty.  Tlie 
first  number  is  three  less  than  the  second,  and  the 
second  is  two  less  than  the  third.  What  are  the 
numbers  ? 

27.  Three    men    together     had    forty-one    dollars. 


08  INTELLECTUAL     ALGEBRA  [^   12 

The  first  had  two  less  than  twice  the  second,  and  th 
third  had  three  more  than  tlie  second.  How  mani 
dollars  had  each  1 

28.  Divide  thirty-eight  into  three  such  parts,  that 
the  first  shall  be  six  less  than  twice  the  second,  and 
the  third  four  less  than  the  second.  What  will  the 
parts  be  ?      ^  ; 

29.  Anna  is  three  years  younger  than  Eliza,  and 
Eliza  is  seven  years  older  than  Lucy-  The  sum  of 
their  ages  is  seventeen.     How  oW  is  each  ? 

30.  Divide  twenty-four  cents  between  three  boys, 
giving  the  first  two  cents  less  than  the  second,  and 
the  second  two  less  than  the  third.  How  many  will 
each  have  ? 

31.  Twenty-six  pupils  in  a  school  are  in  three 
classes.  In  the  first  class  there  are  four  less  than  in 
the  second,  and  in  the  second  three  less  than  in  the 
third.     How  many  pupils  in  each  class? 


SECTION   Xli. 

1.  Robert  and  John  together  have  eleven  cents, 
but  Robert  has  two  more  than  half  as  many  as  John 
How  many  has  each  ? 

Let  X  represent  John's  money  ; 

then 1-2  will  represent  Robert's; 

and  X  -\ [-2  will  represent  the  money  both  had. 


§   12.  INTELLECTUAL     ALGEBRA.  69 

But  both  had  11  cents; 

therefore,  by  the  conditions  of  the  question, 

2;  +  -  +  2=rll. 

Subtracting  2  from  each  member  of  the  equation,  gi^  es 

'     2 

Multiplying  each  member  by  2,  gives 

2x-\-x==  18. 

Uniting  terms  in  the  first  member,  gives 

3x=  18. 

Dividing  each  member  by  3,  gives 

x=zQ  cents,  or  John's  money; 

then  —-{-2  =  5,  Robert's  money. 

2.  Three  boys  together  have  twenty-five  cents. 
William  has  one  fourth  as  many  as  Charles,  and 
George  has  half  as  many  as  Charles  and  four  more. 
How  many  cents  has  each  ? 

3.  Sarah  is  two  years  older  than  Eliza,  and  if  two 
thirds  of  Eliza's  age  be  added  to  Sarah's  age,  the  sum 
will  be  twenty-two.     What  is  the  age  of  each  ? 

4.  What  is  that  number,  to  which  if  you  add  three 
fourths  of  itself  and  five  more,  the  sum  will  be  forty  ? 

5.  If  to  a  boy's  mpney  you  add  three  fifths  of  nis 
money  and  six  cents  more,  the  sum  will  be  fifty-four 
cents.     How  much  money  has  he  ? 

6.  If  three  sevenths  of  a  number  and  nine  more 
be  added  to  the  number  itself,  the  sum  will  be  forty- 
nine.     What  is  the  number  ? 

7.  Three  boys  have  sixty  quills.  The  second  has 
five  more  than  the  third,  and  the  first  has  three  fourths 
as  many  as  the  third.     How  many  has  each? 


70  INTELLECTUAL      ALGEBRA.  [^   12. 

8.  If  you  add  two  fiftlis  of  a  number  and  eight 
more  to  the  number  itself,  the  sum  will  be  fifty. 
What  is  the  number  ? 

9.  A  boy  is  eight  years  older  tlian  his  sister.  If 
twice  the  brother's  age  be  added  to  one  seventh  of  the 
sister's,  the  sum  will  be  thirty-one.     How  old  is  each  1 

10.  The  difference  between  two  numbers  is  three. 
If  half  the  less  be  added  to  three  times  the  greater, 
the  sura  will  be  forty-four.     What  are  the  numbers  ? 

11.  A  farmer  has  five  more  cows  than  horses.  If 
one  third  of  the  number  of  horses  and  two  more  be 
added  to  the  number  of  cows,  the  sum  will  be  twenty 
seven.     How  many  of  each  has  he  ? 

12.  One  number  is  fifteen  more  than  another,  and 
the  sum  of  twice  the  greater,  added  to  three  fourths  of 
the  less,  is  sixty-three.     What  are  the  numbers  ? 

13.  William  is  nine  years  older  than  Robert,  ?n  i 
the  sum  of  twice  ^ViUiam's  age  and  two  fifths  ot 
Robert's,  is  fifty-four.     What  is  the  age  of  each  ? 

14.  What  number  must  be  added  to  three  sevenths 
of  itself  and  five  more,  that  the  sum  may  be  forty- 
five  ? 

15.  A  man  is  seventeen  years  older  than  his  sister. 
If  his  age  be  added  to  three  eighths  of  his  sister's,  the 
sum  will  be  sixty-one.     What  is  the  age  of  each  ? 

16.  The  difference  between  two  numbers  is  ten- 
If  the  greater  be  added  to  three  fourths  of  the  less 
ahd  five  more,  the  sum  will  be  fifty.  What  are  the 
numbers  ? 

3  X 

17.  In  the  equation  a; -}- 10 -| [-7  =  31,  if  the 

terms  in  tlie  first  member  be  united,  and  seventeen 


§13.] 


INTELLECTUAL     ALGEBRA.  71 


subtracted  from  each  member,  what  will  express  the 

^.{uation  ? 

3  X 

18.  If  each  member  of  the  equation,  x-\ =14, 

be  multiplied   by  4,  what  equation  Avill   express  the 
product  1 

19.  If  each  member  of  the  equation  7  a;  =  56,  be 
divided  by  seven,  what  number  will  express  the  value 
of  a;? 

20.  In  the  equation  x-|-7  +  —  -|-3  =  24,  what 
number  will  express  the  value  of  a;  ? 

21.  Reduce  the  equation  2  x  -f"  ~ — h  ^  ^^  25. 
What  will  be  the  value  of  z  ? 

22.  Reduce  the  equation  x-}-3-| \-  -    -\-S:r- 

22.     What  number  does  x  represent  1 


SECTION   XIII. 

1.  George  has  only  four  more  books  than  Anna 
has,  and  yet  he  has  twice  as  many  as  Anna.  How 
many  has  each? 

Let  X  =z  Anna's  books  ; 

then  George  will  have  x-\-A  books. 

But  George  has  also  twice  as  many  as  Annaj 

therefore,  George  has  2  x  books. 

Since  2  x  represents  the  books  George  has,  and  x  -}~  4 

also  represents  his  books,  the  two  expressions 

must  be  equal  to  each  other. 


12  INTELLECTUAL   ALGEBRA.       [§  13 

Therefore,  by  the  conditicms  of  the  question, 

2x  =  x-\-4. 

As  X  and  4  are  equal  to  2  z,  4  alone  must  be  equa. 

to  X  ; 

therefore,  Anna  has  4  books, 

and  George  has  i  -|-  4  r=  8  books  ; 

or,  since  '^  x=z  x-\-4, 

subtracting  x  from  each  member  of  the  equation,  gives 

2  X  —  X  z=  X  —  X  -\-  4. 

Uniting  terms,  gives 

X  =  4,  as  above. 

2.  Robert  is  twice  as  old  as  Charles,  and  the  dif- 
ference between  their  ages  is  five.  What  is  the  age 
of  each  ? 

3.  Anna  had  as  many  dolls  as  Lucy  ;  but  three 
more  were  given  to  Lucy,  and  she  now  has  twice  as 
many  as  Anna.     How  many  dolls  has  each  ? 

Let  X  =.  Anna's  number  of  dolls. 

Lucy  had  the  same ;  tlien  she  also  had  x  dolls, 

and  X  -f-  3  z=  Lucy's  present  number  of  dolls. 

But  this  is  twice  Anna's,  or  2  a; ; 

therefore,  2  x  =  a;  -j-  '^• 

Subtracting  x  from  each  member  of  the  equation,  gives 

x  =  3,  Anna's  number, 

and  x-|-3  =  6,  Lucy's  number. 

4.  To  what  number  must  eight  be  added,  that  the 
sum  may  be  three  times  the  number  ? 

5.  Caroline  is  ten  years  older  than  Mary,  and 
Caroline's  age  is  three  times  Mary's.  How  old  is 
each  ? 

6.  What  number  must  be  increased  by  twelve,  that 
the  sum  may  be  three  times  itself? 


§  13.]  INTELLECTUAL     ALGEBRA.  78 

7.  Lucy  is  fifteen  years  younger  than  Jane,  and 
Jane  is  four  times  as  old  as  Lucy.  What  is  the  age 
of  each  ? 

8.  What  number  will  be  four  times  as  large,  if  you 
add  eighteen  to  it  ?  • 

9.  John  and  James  are  of  the  same  age.  If  to  the 
sum  of  their  ages  twenty-seven  be  added,  the  sum  will 
be  five  times  the  age  of  either.  What  is  the  age  of 
each  ? 

10.  What  number  is  five  less  than  twice  itself? 

11.  What  number  is  that  which  is  eighteen  less 
than  three  times  itself? 

12.  John  has  thirty-three  cents  more  than  Peter, 
and  he  has  four  times  as  much  money  as  Peter.  How 
many  cents  has  each  ? 

13.  What  number  must  be  added  to  itself,  and  then 
be  increased  by  five,  that  the  sum  may  be  three  times 
tiie  number  ? 

14.  A  boy  is  four  times  as  old  as  his  playmate  ; 
and  Ihe  difference  in  their  ages  is  twelve  years.  What 
is  the  age  of  each  ? 

15.  The  difference  between  two  numbers  is  three, 
and  twice  the  larger  is  equal  to  three  times  the  small- 
er.     What  are  the  numbers  ? 

16.  A  father  is  five  times  as  old  as  his  son,  and  the 
difference  of  their  ages  is  thirty-two.  V/hat  is  the 
age  of  each  ? 

17.  One  number  is  six  times  another,  and  the  dif- 
ference between  them  is  forty.    What  are  the  numbers? 

18.  If  you  give  Sarah  sixteen  books  more  than  you 
give  Caroline,  she  will  have  three  times  as  many  as 
Caroline.     How  many  books  will  each  have  ? 


74  INTELLECTUAL     ALGEBRA.  [§  13. 

19.  If  forty  years  be  added  to  a  person's  age,  the 
sum  will  be  three  times  his  age.     AV'hat  is  his  age  ? 

20.  The  dillerence  between  two  numbers  is  four, 
and  three  times  the  larger  is  equal  to  five  times  the 
smaller.     What  are  the  numbers  ? 

21.  If  Charles  lives  seven  years  longer,  his  age 
will  be  double  what  it  now  is.     What  is  his  age  ? 

22.  If  twelve  be  added  to  some  number,  the  sum 
will  be  four  times  that  number.     What  is  the  number  ? 

23.  If  John  lives  fourteen  years  more,  he  will  be 
three  times  as  old  as  he  is  now.     How  old  is  he? 

24.  If  twenty  cents  be  added  to  a  boy's  money,  the 
sum  will  be  three  times  as  much  as  it  now  is.  How 
much  money  has  he  ? 

25.  Fifteen  years  hence,  Mary's  age  will  be  four 
times  what  it  now  is.     What  is  her  age  ? 

20.  If  twenty  be  added  to  some  number,  the  sum 
will  be  five  times  that  number.     What  is  the  number? 

27.  If  you  give  twenty-four  cents  more  to  James, 
he  will  have  five  times  as  many  as  he  now  has.  How 
many  has  he  ? 

28.  What  number  must  be  added  to  itself  and  to 
seven  more,  that  the  sum  may  be  three  times  that 
number  ? 

29.  When  George  shall  be  thirty  years  older,  his 
age  will  be  four  times  as  much  as  it  is  now.  What  is 
his  age  ? 

30.  What  number  must  be  added  to  itself  and  to 
twenty-seven  more,  that  the  sum  may  be  five   iimes 

hat  number  ? 
33.    John  is  four  years  older  than  Hem  v,  and  twice 


[§  13.  INTELLECTUAL     ALGEBRA.  75t. 

John's  age  is  three  tiirtes  Henry's  age.      What   is  the 
age  of  each '? 

32.  The  difference  between  two  numbers  is  ten, 
and  three  times  the  'greater  is  equal  to  five  times  the 
less  number.     What  are  the  numbers  1 

33.  One  nian  is  twelve  years  older  than  his  brother, 
and  three  times  the  man's  age  is  equal  to  five  times 
the  brother's.     How  old  is  each  ? 

34.  The  difference  of  two  numbers  is  six,  and  five 
times  the  greater  is  equal  to  eight  times  the  less. 
What  are  the  numbers  ? 

35.  A  man  is  six  years  older  than  his  wife,  and 
seven  times  the  man's  age  is  equal  to  nine  times  his. 
wife's.     WJiat  is  the  age  of  each? 

36.  One  number  is  nine  more  than  another,  and 
four  times  the  greater  is  seven  times  the  less.  What 
are  the  numbers  ? 

37.  If  x-}-G  be  multiplied  by  3,  what  will  express 
the  product? 

38.  If  2  a;  -(-  4  be  multiplied  by  5,  what  will  ex- 
press the  product  ? 

39.  In  the  equation  3  z  -{-  12  =r:  5  r,  if  3  a;  be  sub- 
tracted from  each  member,  what  will  express  the  re- 
sult? 

40.  If  each  member  of  the  equation  2  a;  =  12  be 
divided  by  2,  what  equation  will  express  the  quotient? 

41.  In  the  equation  2  r -|- 15  =  7  x,  what  number 
will  express  the  value  of  a:  ? 

42.  Reduce  the  equation  3x-f-9  =  6x-|-3.  What 
number  does  x  represent  ? 

43.  Reduce  the  equation  r-\-20z=zQ4.  What  ia 
the  value  of  x  1 


ib  INTELLECTUAL  ALGEBRA.       [§  14 

/ 

SECTION   XIV. 

I.  Eliza  jfave  half  of  her  books  to  Joshua,  and  one 
fourth  of  them  to  Samuel,  and  then  had  but  two  left 
for  herself  How  many  had  she  at  first,  and  how 
many  did  she  give  to  each? 

Let  X  =:  the  whole  number  of  books  ; 

then  —  :=:  the  books  she  gave  to  Joshua, 

and  —  =z  the  books  she  gave  to  Samuel ; 

then \-  - — [-2  must  represent  all  the  books  she  had. 

Therefore,  by  the  conditions  of  the  qu^tion. 

Multiplying  each  member  by  4,  gives 

lf_Li:f_i_8  =  4z. 

2  4 

Reducing  fractions  in  the  first  member,  gives 

2x-\-x-\-S  =  4:X; 

uniting  terms  in  the  first  member, 

3  a;  +  8  —  4  X. 

Subtracting  3  x  from  each  member,  gives 

8  —  x. 

Therefore  Eliza  had  ,8  bboks  ; 

then  —  =  4,  Joshua's  books  ; 

2 

and  ~=z2,  Samuel's  books. 
4 

2.    Sarah  gave  one  third  of  her  money  for  a  book, 

and  one  sixth  of  it  for  a  pencil,  and  then  had  eight 

sents  left.     How  many  cents  had  she  at  first  f 


§   14.]  1NTELL.ECTUA].      ALGEBKA.  77 

3.  If  one  half  of  a  number  be  added  to  one  third 
of  the  same  number  and  four  more,  the  sum  will  be 
the  number  itself.     What  is  the  number  1  ■ 

4.  A  man,  being  asked  his  age,  replied,  "If  ten 
years  be  added  to  one  fourth  and  one  third  of  my  age, 
the  sum  will  be  my  age."     How  old  was  he  1 

5.  If  two  thirds  of  a  number  and  seven  more  be 
added  to  one  sixth  of  the  same  number,  the  sum  will 
be  equal  to  the  number.     What  is  the  number  1 

6.  Daniel  gave  one  fourth  of  his  cherries  to  one 
boy,  and  one  sixth  of  them  to  another,  and  had  twen- 
ty-one left.     How  many  had  he  at  first  ? 

7.  What  number  is  that  which  is  equal  to  the  sum 
of  two  thirds  and  one  fourth  of  itself  added  to  five  ? 

8.  A  farmer  kept  his  cows  in  three  pastures.  In 
one  pasture  he  had  one  tliird  of  them,  in  another  one 
fourth  of  them,  and  ten  cows  in  the  third  pasture. 
How  many  cows  had  he  1 

9."  If  one  fourth  and  one  sixth  of  a  number  be 
added  to  thirty-five,  the  sum  will  be  the  number  it- 
self    What  is  the  number  ? 

10.  Thomas  had  one  fourth  of  his  money  in  a 
purse,  three  eighths  of  it  in  his  pocket,  and  nine  cents 
in  his  hand.     How  many  cents  had  he  ? 

11.  The  sum  of  two  fifths  and  three  tenths  of  a 
number,  is  twelve  less  than  the  number  itself.  What 
is  the  number  ? 

12.  In  an  orchard,  one  fourth  of  the  trees  are  plum- 
trees  one  third  cherry-trees,  and  fifteen  are  pear-trees. 
How  many  trees  are  there  in  the  orchard  ? 

13.  What  number  is  that  which  is  equal  to  the  sum 
f  one  half  and  one  fifth  of  itself  added  to  eighteen  ? 


78  INTELLECTUAL     ALGEBRA.  §   14.  J 

14.  A  man  broke  one  fifth  of  his  eggs  on  his  way 
to  market ;  while  there  he  sold  three  tenths  of  them, 
and  still  had  twenty  left.  How  many  eggs  did  he 
have  when  he  started  from  home  ? 

15.  If  two  thirds  and  one  ninth  of  a  number  be 
added  to  sixteen,  the  sum  will  be  the  number  itself 
What  is  the  number  ? 

16.  One  sixth  of  the  pupils  of  a  school  are  in  the 
first  class,  one  fourth  in  the  second,  one  third  of  them 
in  the  third,  and  twenty-four  in  the  fourth  class.  How 
many  pupils  are  there  in  the  school  ? 

17.  The  sum  of  one  half,  one  fifth,  and  one  tenth 
of  a  number,  is  eight  less  than  the  number  itself. 
What  is  the  number  ? 

18.  One  third  of  a  man's  farm  is  used  as  a  pasture 
for  his  cattle,  one  sixth  is  meadow,  one  twelfth  wood- 
land, and  the  remaining  twenty  acres  he  cultivates 
with  the  plough.     How  many  acres  has  he  ? 

19.  If  one  third,  one  sixth,  and  one  ninth  bf  a 
number  be  added  to  fourteen,  the  sum  will  be  the 
number  itself     What  is  the  number  ? 

20.  In  a  college  one  fifth  of  the  students  are 
seniors,  one  fourth  juniors,  three  tenths  sophomores, 
and  there  are  forty  students  in  the  freshmen  class. 
How  many  students  are  there  in  the  college  ? 

21.  Reduce    the   expression 1 1-- —      What 

^  2  4  8 

term  will  express  the  sum  ? 

22.  Reduce  the  expression  ~-  ~\ — ^  -f-  T*  What 
will  express  the  sum  ? 

23.  If  each  member  of  the  equation  — ;^  -| \ — 


(^  15.]  INTELLECTUAL.     ALGEBRA.  7'^ 

=  13  be  multiplied  by  12,  what  will  express  the  equa- 
tion before  it  is  reduced  ? 

24.  If  the  terms  in  the  first  member  of  the  equa- 

24  X         12. T         12  .r  ^ 

cion -{ ^  z=  15G,  be  reduced  to  one  term, 

3     '      4     '      6  ' 

what  will  the  equation  be  ? 

25.  If  each  member  of  the  equation  13  x  :r=  1,56  be 
divided  by  13,  what  number  will  show  the  value  of  x? 

(■         3  X 

26.  Reduce  the  equation  —  4" 'T"f~'''—1®-   What 
number  does  x  represent  ? 

27.  Reduce  the  equation 1 \-^-\-5  =  22. 

What  number  will  express  the  value  of  x  ? 


SECTION    XV. 

1.    Charles  gave  away  one  fifth  of  his  money,  and 
had  eight  cents  left.     How  many  cents  had  he  at  first? 
Let  X  =  the  money  Charles  had  ; 

then  —  ^  the  money  he  gave  away. 

But  he  had  8  cents  left ; 
therefore,  by  the  conditions  of  the  question, 

x  —  —  =  S. 

5 

Multiplying  each  member  of  the  equation  by  5,  gives 

5  z  —  a;  =:  40  ; 

uniting  terms  in  the  first  member, 

4z  =  40; 

dividing  each  member  of  the  equation  by  4, 

x=zlO  cents,  the  money  Charles  had. 


80  INTELLECTUAL     ALGEBRA  [§   15, 

9.  A  boy  spent  one  fifth  of  his  money,  and  gave 
away  two  fifths  of  it.  He  then  had  six  cents  left 
How  many  cents  had 'he  at  first? 

T,et  x^^the  money  he  had  at  first ; 
then  —  i=z  the  money  he  spent, 

2  X 

and  -^  =  what  he  gave  away  ; 

X         2  X 
then  X —  must  be  equal  to  what  he  had  left 

But  he  had  6  cents  left ; 

therefore,  x  — ^^=16. 

0         5 

bince3:  =  — ,  then =  6. 

3  5  5  5 

Uniting  terms  in  the  first  member,  gives 

2  X 

5 
If  f  of  x=zij,  Aof  a;  =  ^of  6. 

[f  i  of  3;  =  3,  the  whole  of  x  =  15  cents,  the  boy's 
money. 

Or, 

2x 

Since  —  =  0, 
5 

dividing  each  member  of  the  equation  by  2,  gives 

5 
Multiplying  each  member  by  5,  gives 
X  =  15,  as  above. 

Remark.  —  Form  tlie  equations  in  this  section  by  subtract- 
ing the  parts,  «S:c. 

r»?^che.s.   and   gave 


i^  15.3  INTELLECTUAL     ALGEBRA.  &% 

away  one  third  of  them.     She  had  five  left.     How 
many  had  she  at  first? 

4.  If  from  some  number  you  subtract  one  half  of 
itself  and  one  third  of  itself,  the  remainder  wi!.l  be 
seven.     What  is  the  number  1 

5.  A  boy  spent  one  half  of  his  money  for  writing- 
books,  and  one  sixth  of  it  for  pencils.  He  had  twelve 
cents  left.     How  much  money  had  he  at  first  ? 

6.  From  what  number  must  one  third  and  one 
sixth  of  the  same  number  be  taken,  that  the  remain- 
der may  be  eighteen  ? 

7.  Mary  lost  one  third  of  her  money,  and  gave 
away  two  fifths  of  it.  She  had  eight  cents  left.  How 
many  cents  had  she  at  first  1 

8.  From  what  number  must  one  half  and  one  tenth 
of  itself  be  taken,  that  sixteen  may  be  left  1 

9.  A  man  paid  one  fourth  of  his  money  for  a  pair 
of  boots,  and  one  fifth  of  it  for  a  hat.  He  had  eleven 
dollars  left.  How  much  money  had  he  at  first?  ann 
how  much  did  he  pay  for  each  of  his  purthases  ? 

10.  A  boy  spent  one  half  of  his  money,  and  gave 
away  one  fifth  of  it.  He  then  had  twelve  cents  left 
How  many  cents  had  he  at  first? 

11.  If  from  a  number. you  subtract  one  fourth  ot 
itself  and  one  third  of  itself,  the  remainder  will  bt 
ten.     What  is  the  number  ? 

12.  If  from  a  man's  age  one  fourth  and  one  sixtL 
of  his  age  be  subtracted,  the  remainder  will  be  four 
teen  years.     What  is  the  man's  age  1 

13.  From  what  number  must  you  take  two  fifths 
and  three  fifteenths  of  itself,  that  the  remainder  maj 

'  be  twenty-four  ? 
6 


82  INTELLECTUAL     ALGEBRA.  [§  15. 

14.  A  farmer  soid  one  fourth  of  his  flock  of  sheep 
to  one  man,  and  two  fifths  to  another,  and  he  still  had 
fifty-six  sheep  left.  How  many  sheep  were  in  the 
flock  at  first  ? 

15.  If  from  some  number  you  subtract  one  third 
and  four  ninths  of  itself,  the  remainder  will  be  four- 
teen.    What  is  the  number  ? 

10.  A  man  sold  one  third  of  his  farm  to  one  per- 
son, and  two  ninths  of  it  to  another,  retaining  only 
forty-eight  acres  for  himself  Of  how  many  acres  did 
the  farm  at  first  consist  1 

17.  From  what  number  must  one  half  and  tliree 
fourteenths  of  the  same  number  be  taken,  that  the 
remainder  may  be  eight  ? 

18.  A  boy  gave  one  half  of  his  chestnut?  to  one 
playmate,  one  fifth  to  another,  and  one  tenth  to  a 
third,  and  kept  twenty-four  for  himself  How  many 
had  he  at  first  ? 

19.  If  one  fourth,  one  sixth,  and  one  twelfth  of  a 
number  be  subtracted  from  itself,  the  remainder  will 
be  forty-eight.     What  is  the  number  ? 

20.  A  drover  on  his  way  sold  one  third  of  his  cattle 
to  one  man,  and  one  ninth  to  another,  and  then  had 
fifty  left.     How  many  were  in  the  drove  at  first  ? 

21.  If  from  five  fifths,  or  the  whole  of  x,  two  fifths 
of  X  be  subtracted,  what  wUl  remain  ? 

•  22.    If  from  x,  two  fifths  of  x  and  one  fifth  of  x  be 
subtracted,  what  will  express  the  remainder  ? 

23.  If  from  six  sixths,  or  the  whole  of  x,  one  half 
of  X  and  one  third  of  x  be  subtracted,  what  will  rep 
resent  the  remainder  ? 


§  15.]  INTELLECTUAL     ALGEBRA-  @ig| 

24.  Reduce    the    expression    x .     What 

will  represent  it  ? 

2  X 

25.  What  will  represent  the  expression  x 

— ,  when  reduced  ? 
3' 

X  ~  X 

26.  What  will  the  expression  x become 

when  it  is  reduced  ? 

24  X 

27.  If  from  ~ — ,  or  twice  the  whole  of  z,  \i  of  x 

12  '  '  ^- 

be  taken,  what  w'ill  represent  the  remainder  ? 

28.  What  will  the  expression  2  a; 


2  3  4 

become  when  it  is  reduced  1 

29.  Reduce  the  expression  2x ^ .     What 

will  represent  it  ? 

30.  In  the  equation  x ^  rr:  2,  if  each  term 

of  the  first  member  be  changed  to  sixths,  what  will 
the  equation  be  ? 

6  a;        3  a;        2  a; 

31.  In  the  equation ^^=z2,  where  the 

^  6  6  6' 

fractions  are  reduced  to  the  same  denominator ,  if  the 
terms  of  the  first  member  be  united,  what  will  express 
the  equation  ? 

32.  In   the   equation  —  =  2^  if   each   member   be 

6 

multiplied  by  the  denominator,  6,  the  equation  will  be 
cleared  of  fractions.     What  will  be  the  value  oi  x1 

33.  If  the   fractions  —  and  —  be  reduced  to  the 

3  4 

same  denominator,  what  will  be  the  common  denomina- 
tor f  and  what  will  the  fractions  be  ? 


84  INTF.LLCCTfJAL,     ALGEBRA.  [§   1^" 

34.  Reduce  the  equation  x r=:  ().     What 

^  2  5 

is  the  number  represented  by  x  ? 

X         3  a-" 

35.  If  the  equation  2x  —  — =  9  be  reduced, 

what  will  be  the  value  of  x  ? 

36.  In  the  equation  2x  —  — ^  =  7,  if  all  the 

terms  of  the  first  member  be  reduced  to  a  common 

20  X       5  X        8  X 

denominator,  the  equation  will  be ^7. 

'  ^  10  10         10 

If  this  equation  be  reduced,  what  will  be  the  value 

of  X  1 


SECTION   XVI. 

1.    If  Charles  eats  one  third  of  a  barrel  of  apples  in 
a  week,  in  how  many  weeks  will  he  eat  three  thirds  or 
the  whole  barrel  ? 
Let  X  represent  the  time,  or  number  of  weeks,  in  which 

he  will  eat  tlie  whole  barrel. 
If  he  eats  -|  of  a  barrel  in  one  week,  in  x  weeks  he 

will  eat  —  of  a  barrel. 
3 

But  in  X  weeks  he  eats  the  whole  barrel  ; 

therefore,  —  of  a  barrel  must  be  equal  to  the  whole 
'     3  ' 

barrel  ; 

.T  3 

that  IS,  —  rr:  —  or  a  whole  one. 
3  3 

If  ^  of  X  r=  ^  of  3,  the  whole  of  x  will  be  equal  to 

the  whole  of  3 ; 


vj  16  ]       INTELLECTUAL  ALGEBRA.  85 

therefore,  x^—S, 

and  lie  will  eat  the  barrel  in  3  weeks. 

Or, 

by  the  conditions  of  the  question,  —=r} 

3 

Multiplying  by  3,  gives  z  m  3. 

'2.  If  George  eats  one  sixth  of  a  barrel  of  apples 
in  one  week,  how  long  will  six  sixths,  or  a  whole  bar- 
rel, last  him  ? 

3.  George  eats  one  sixth  of  a  barrel  of  apples  in  a 
week,  and  Charles  eats  one  third  of  a  barrel  in  the 
same  time.  In  how  many  weeks  will  both  together 
eat  the  whole  barrel  ? 

Let  X  =  the  number  of  weeks  in  which  both  will  eat 
a  whole  barrel. 

George,  in  x  weeks,  eats  -^  of  the  barrel ; 

6    J  '6 

Charles,  in  x  weeks,  eats  —  of  the  barrel  ; 

both  toorether,  in  x  weeks,  eat 1-  —  of  the  barrel : 

°  '  .  3     '     6 

But  both,  in  x  weeks,  eat  the  whole  barrel ; 

therefore,  by  the  conditions  of  the  question, 

3*6 
and  reducing  the  fractions  to  the  same  denomination, 

gives 

6     '     6 

Clearing  the  equation  from  fractions,  by  multiplying 

by  G,  the  denominator,  gives 

2x-\-x  =  6; 

uniting  terms  in  the  first  member, 

3  X-  =  6 


B6  INTELLECTUAL,  ALGEBRA.        [§  16 

Dividing  by  3,  gives  x^2; 

therefore,  both  together  eat  the  barrel  in  2  weeks. 

4.  How  many  times  is  the  sum  of  one  sixth  and 
one  third  contained  in  one? 

•     5.    A  man  can  dig  one  fourth  of  a  trench  in  one 
week      In  how  many  weeks  can  he  dig  the  whole  of  it  ? 

6.  A  man  can  build  one  sixth  of  a  stone  wall  in  one 
day.     In  how  many  days  will  he  build  the  whole  wall? 

7.  A  man  can  build  one  third  of  a  stone  wall  in 
one  day.  How  many  days  will  it  take  him  to  build 
the  whole  of  it  1 

8.  One  man  can  build  one  third  of  a  wall  in  one 
day,  and  another  man  can  build  only  one  sixth  of  the 
same  wall  in  a  day.  In  how  many  days  will  both  men, 
working  together,  build  the  wall  ? 

9.  If  a  man  can  eat  one  eighth  of  a  barrel  of  bread 
in  a  week,  how  long  will  a  whole  barrel  last  him? 

10.  If  a  man  can  do  two  ninths  of  some  stated 
piece  of  work  in  one  day,  how. many  days  must  he 
work  to  do  the  whole  of  it? 

Let  X  represenf'the  number  of  days  in  which  he  can 

do  the  whole  of  it. 

Since  he  does  f  of  it  in  .one  day,  in  x  days  he  will  do 

2  X 
X  times  two  ninths  of  it,  which  is  — ^. 

"      9 

But  in  X  days  he  will  do  all  of  it ; 

therefore,  ^^  must  be  equal  to  the  whole  of  it,  or  1  ; 
,        .      2x        , 

that  IS,  —  r=  1. 
'    9 

If  ^  of  2  a;  =  1,  the  whole  of  2  x  =  9  times  1. 

If  2  z  =  9,  X  =  f ,  which  is  4*  ; 

therefore,  he  would  do  the  work  in  4J  days. 


<J   It). J  INTELLECTUAL     ALGEBRA  87 

11.  A  man  can  shingle  one  fourth  of  the  roof  of  a 
house  in  one  day,  and  a  boy  can  shingle  one  twelfth 
of  it  in  a  day.  How  many  days  will  it  take  for  both, 
working  together,  to  shingle  the  roof? 

12.  How  many  times  the  sum  of  one  fourth  and 
one  eighth  of  any  thing  will  it  take  to  make  the  whole 
of  the  same  thing  ? 

13.  How  many  times  is  the  sum  of  one  fourth 
and  one  twelfth  of  any  thing  contained  in  the  whole 
of  it  ? 

14.  One  horse  eats  one  sixth  of  a  ton  of  hay  in  a 
week,  and  another  horse  eats  only  one  twelfth  of  a 
ton  in  the  same  time.  How  long  will  a  ton  of  hay 
last  both  of  them  ? 

15.  How  many  times  is  the  sum  of  one  sixth  and 
one  twelfth  contained  in  one  ? 

16.  How  many  times  is  the  sum  of  one  third  and 
one  twelfth  contained  in  two  ? 

17.  A  man  can  do,  in  one  week,  one  fourth  of  the 
work  required  to  repair  a  house,  and  a  boy  can  do 
three  sixteenths  of  the  same  piece  of  work  in  the 
same  time.  In  how  many  weeks  will  both,  working 
together,  do  it  1 

18.  How  many  times  is  the  sum  of  one  third  and 
two  sevenths  contained  in  one  1 

1)9.  How  many  times  is  the  sum  of  two  thirds  and 
five  twelfths  contained  in  three  ? 

20.  Two  men  and  a  boy  are  employed  to  make  a 
fence.  One  man  can  do  o\\e  fourth  of  it  in  a  day, 
the  other  can  do  one  fifth  of  it  in  the  same  time,  and 
the  boy  can  do  one  twentieth  of  it  in  a  day.  How 
many  days  will  it  take  them  all  to  do  it  ? 


88  INTELLKCTUAL     ALGEBRA.  §   16.] 

21.'  How  many  times  is  the  sum  of  one  fourth,  one 
sixth,  and  one  twelfth,  contained  in  a  whole  one? 

2'2.  How  many  times  is  the  sum  of  three  fourths, 
five  sixths,  and  seven  twelfths,  contained  in  three  ? 

23.  A  father  and  his  son  have  but  one  barrel  of 
bread.  The  father  eats  three  twentieths  of  a  barrel 
in  a  week,  and  the  son  one  tenth  of  a  barrel  in  the 
same  time.     How  long  will  it  last  them  ? 

24.  How  many  times  the  sum  of  one  third,  one 
seventh,  and  one  twenty-first,  will  it  take  to  make  a 
whole  one  1 

25.  One  man  can  do  one  third  of  a  given  piece  of 
work  in  one  day  ;  another  can  do  one  eighth  of  the 
same  work  in  a  day ;  and  a  boy  can  do  one  twenty- 
fourth  of  it  in  the  same  time.  How  many  days  will 
it  take  the  three,  working  together,  to  get  it  done  ? 

26.  Reduce   — ,  — ,  and  —  to  the  same  denomi- 

.37  21 

nation,  and  what  will  they  become  ? 

27.  What  will  express  the  sum,  if 1-  — — j 

be  reduced  to  one  term? 

28.   Reduce  the  equation -^ — \- - — I-  — =1.    What 
^  -1     '     li     '    12 

number  does  j  represent? 

2  X  X 

29.  If  the  equation  ^^- -| ::=  1,  be  reduced,  what 

will  express  the  value  of  x  ? 

30.  Reduce  the  equation —4--- 4--^  =  1.    What 

*  3  .^  4    "^  6 

will  be  the  number  reprcL-ented  by  x  1 


^5  17.  I  INTELLECTUAL     ALGEBRA.  8^ 


SECTION   XVII. 

1.  If  from  two  thirds  of  Catherine's  age  you  suo- 
tract  one  third  of  her  age,  the  difference  will  be  four 
years.     How  old  is  she  ? 

Let  X  represent  her  age ; 
then,  by  the  conditions  of  the  question 

3        ~3'~ 
But  —  less  —  =  —  ; 

3  3  3 

therefore,  —  =  4. 

'    3 
If  •^-  of  X  m  4,  the  whole  of  x  must  be  3  times  4  ; 
then  X  =z  12,  or  Catherine's  age. 

2.  If  from  one  half  of  a  boy's  money  one  fourth 
of  his  money  be  taken,  three  cents  will  remain.  How 
many  cents  has  he  ? 

3.  The  difference  between  one  half  and  one  fourth 
of  the  same  number  is  five.     What  is  the  number  ? 

4.  If  from  one  third  of  uncle  William's  age  one 
fifth  of  his  age  be  taken,  the  difference  will  be  eight 
years.     What  is  his  age  ? 

5.  If  two  fifths  of  some  number  be  taken  from 
seven,  tenths  of  the  same  number,  the  difference  will 
be  twelve.     What  is  the  number  ? 

6.  Mary's  age  is  two  thirds  of  her  brother's  age, 
and  Jane's  is  four  ninths  of  the  same  brother  s  age. 
The  difference  between  Mary's  age  3nd  Jane's,  is 
eight  years.     How  old  is  the  brother,  and  each  sister  1 


90  INTELLECTUAL  ALGEBRA.       [§  17 

7.  The  difference  between  two  thirds  and  four 
sevenths  of  the  same  number  is  four.  What  is  the 
number  ? 

8.  If  three  fourteenths  of  some  number  be  taken 
from  two  sevenths  of  the  same  number,  the  remainder 
will  be  two.     What  is  the  number  ? 

9.  A  boy  eat  one  fourth  of  his  plums,  and  gave 
away  one  fifth  of  them.  The  difference  between  what 
he  eat  and  what  he  gave  away  was  three.  How  many 
had  he?  and  how  many  did  he  give  away  ? 

10.  If  three  eighths  of  some  number  be  taken  from 
three  fourths  of  the  same  number,  the  remainder  will 
be  six.     What  is  the  number  ? 

11.  If  from  half  of  a  man's  money  one  seventh 
of  his  money  be  taken,  the  difference  will  be  fifteen 
dollars.     How  many  dollars  has  he  ? 

12.  The  difference  between  three  fourths  and  five 
sixths  of  the  same  number  is  nine.  What  is  the 
number  1 

13.  A  man  owned  seven  tenths  of  a  flock  of  sheep. 
After  selling  two  fifths  of  the  whole  flock,  he  had 
thirty  sheep  still  belonging  to  him.  How  many  sheep 
were  in  the  flock  before  the  sale  ? 

14.  If  two  sevenths  of  some  number  be  taken  from 
one  half  of  the  same  number,  the  difference  between 
the  two  parts  of  it  will  be  twelve.  What  is  the  whole 
number  ? 

15.  John's  money  is  two  thirds  of  William's  money, 
and  Henry's  is  one  ninth  of  William's.  The  differ- 
ence between  John's  money  and  Henry's  is  fitlLeen 
cents.     How  much  money  has  each  ? 

10.   The  difference  between  three  fourth*;  and  three 


[^   17.  INTELLECTUAL     ALGEBRA.  91 

fifths  of  the    same   number  is  twelve.     What  is  the 
number  ? 

17.  John  owned  two  thirds  of  a  basket  of  eggs, 
and,  after  selling  one  fifth  of  all  there  were  in  the 
basket,  fourteen  eggs  still  belonged  to  him.  How 
many  were  in  the  basket  at  first  ? 

18.  If  from  two  thirds  of  some  number  three 
sevenths  of  the  same  number  be  subtracted,  the  dif- 
ference will  be  fifteen.     What  is  the  number  ? 

19.  Two  fifths  of  a  pole  are  in  the  water,  one  tenth 
m  the- mud,  and  the  remainder  out  of  water.  There 
are  nine  feet  more  of  it  in  the  water  than  in  the  mud. 
How  long  is  the  pole  ? 

20.  The  difference  between  two  thirds  and  two 
nmths  of  the  same  number  is  thirty-six.  What  is 
the  number  1 

21.  Three  sevenths  of  a  flock  of  sheep  were  put 
in  one  pasture,  three  fourteenths  in  another,  and  the 
rest  were  sold.  There  were  twelve  more  sheep  in  one 
pasture  than  in  the  other.  Of  how  many  sheep  did 
the  flock  consist  ?  how  many  were  in  each  pasture  ? 
and  how  many  were  sold  1 

22.  The  difference  between  three  fourths  and  seven 
eighths  of  the  same  number  is  eleven.  What  is  the 
number  ? 

23.  If  seven  twelfths  of  some  number  be  taken 
from  the  same  number,  the  difference  will  be  thirty. 
What  is  the  number  ? 

24.  If  the  terms  in  the  first  member  of  the  equation 

—  —  —  =:10,  be  reduced  to  the  same  denomination, 
4         8  ' 

and  then  to  one  term,  what  will  the  equation  be? 


92  INTKLLECTUAL     ALGEBRA.  [§  18. 

25.  In  the  equation  —  :=  10,  what  number  is  rep- 
resented by  I  ? 

26.  Reduce   the   equation =zlo       What 

number  will  express  the  value  of  xl 

3  X        2  X 

27.  If  the  equation =  1  be  reduced,  vhat 

will  be  the  value  of  a;  ? 

28.  Reduce    the    equation r  =  ^-       What 

number  does  x  represent  ? 

29.  What  nuiriber  will  express  the  value  of  x  in 

the  equation ^rrr20? 

^  3  G 

30.  What  is  the  number  represented  by  x  in  the 
equation r=  9  ? 

^  2    .       20 


SECTION   XVIII. 

1.  The  sum  of  the  ages  of  two  boys  is  twelve 
years,  and  the  elder  is  twice  the  age  of  the  younger 
What  is  the  age  of  each  ? 

Let  X  represent  the  age  of  the  younger  boy; 

then  12  —  x  will  express  the  age  of  the  elder, 

and  12  —  x  must  be  equal  to  twice  z. 

Then,  by  the  conditions  of  the  question, 

12  — x  =  2r. 
If  12,  with  X  taken  from  it,  is  equal  to  2x, 
12  without  x  taken  from   it,  must   be   equal  to  one 
.  •  more  x; 


^  18.]  INTELLECTUAL     ALGEBKA.  93| 

therefore,  12  =  3  x.  and  4  =  a;. 

Then  the  age  of  the  younger  boy  is  4  years  ^ 

and  12  —  z  =  8,  the  age  of  the  elder. 

Or, 

since  12  —  xz=z2x, 

adding  x  to  each  member  of  the  equation,  gives 

12  —  x-{-x=z'^x-\-x. 

Uniting  terms  in  each  member,  gives 

12  =  '3x. 

Dividing  each  member  by  3,  gives 

4  =  X,  as  above. 

2.  The  sum  of  two  numbers  is  fifteen,  and  the 
greater  is  twice  the  .smaller.  What  are  the  num- 
bers ? 

3.  If  12  —  X  be  added  to  12  —  x,  what  will  express 
the  sum  ? 

It  may  be  expressed  thus  ;  12  —  x  -{-  12  —  x. 
Uniting  terms,  24  —  2  x,  Ans. 

4.  If  12  —  X  be  multiplied  by  2,  what  will  express 
the  product  ? 

5.  If  12  —  X  be  multiplied  by  5,  what  will  the 
product  be  ? 

6.  George  and  Anna  together  have  ten  books,  and 
twice  George's  books  are  equal  to  three  times  Anna's. 
How  many  has  each  1 

Let  X  represent  Anna's  books ; 

then  10  —  X  will  represent  George's  books  ; 

twice  George's  books  will  be  twice  10  —  x,  which  is 

20  — 2x> 

But  twice  George's  are  equal  to  three  times  Anna's ; 

therefore,  by  the  conditions  of  the  question 

20  — 2x=:3x. 


94  INTELLECTUAL     ALGEBRA.  [^18 

Adding  2x  to  each  member  of  the  equation,  gives 

Uniting  terms  in  each  member,  gives 

20  =  0  3:. 

Dividing  each  member  by  5,  gives 

,       4  =  x; 

therefore,  Anna  has  4  books , 

and  10  —  x  =  6; 

then  George  has  G  books. 

7.  The  sum  of  two  numbers  is  twenty,  and  twice 
the  greater  is  three  times  the  other.  What  are  the 
numbers  1 

8.  The  sum  of  the  ages  of  two  boys  is  eighteen 
years,  and  four  times  the  age  of  the  younger  boy  will  be 
twice  the  age  of  the  other.     What  is  the  age  of  each  ? 

9.  Twice  one  number  is  four  times  another,  and 
their  sum  is  thirty.     What  are  the  numbers  1 

10.  Two  men  are  to  share  thirty-five  dollars  be- 
tween them,  and  A  is  to  have  four  times  as  L;nany  as 
B.     How  many  dollars  will  each  have  ? 

Let  X  represent  B's  share  ; 
then  35  —  x  will  express  A's. 

11.  The  sum  of  two  numbers  is  forty-two,  and 
three  times  one  number  is  four  times  the  other. 
What  are  the  numbers  ? 

12.  A  farmer  has  seventy  sheep  in  two  pastures, 
and  twice  the  number  of  sheep  in  the  larger  pasture 
will  be  equal  to  five  times  the  sheep  in  the  other. 
How  many  sheep  in  each  pasture  ? 

13.  Three  times  one  number  Is  five  times  a  smaller, 
and  their  sum  is  twenty-four.    What  are  the  numbers  1 

14    George  had  seventy-two  cents,  and  lost  a  part 


§  18.]  INTELLKCTUAI,     ALGEBRA.  96 

of  them.  Three  times  the  number  he  lost  is  equal  to 
six  times  the  number  he  had  left.  How  many  cents 
did  he  lose  ? 

15.  The  sum  of  two  numbers  is  seventeen,  and 
twice  the  greater  is  two  more  than  three  times  the 
less.     What  are  the  numbers  ? 

'  Let  X  represent  the  less  number  ; 

then  16- — x  will  represent  the  greater. 

Twice  the  greater  is  32  —  2x,  which  is  2  more  than 

3t. 

Therefore,  by  the  conditions  of  the  question, 

32  — 2a;  — 2  =  3x; 

and,  uniting  terms  in  the  first  member, 

30  — 22;  =  33:. 

Adding  2  x  to  each  member  of  the  equation,  gives 

30— 2z-|-2x=:32;  +  2x. 

Uniting  terms,  gives 

30  rr  5  a:. 

Dividing  each  member  by  5,  gives 

6  =  2;; 

therefore,  6  is  the  smaller  number, 

and  16  —  %-=i  10,  the  larger  number. 

16.  Two  men  together  spend  one  hundred  dollars. 
Three  times  the  money  A  spends  is  equal  to  seven 
times  the  money  B  spends.  How  many  dollars  does 
each  spend  ? 

17.  The  sum  of  two  numbers  is  twenty-eight,  and 
the  greater  is  four  less  than  three  times  the  smaller. 
What  are  the  numbers? 

18.  Divide  thirty-one  into  two  such  parts,  that 
twice  the  greater  shall  be  two  more  than  thiee  times 
the  less.     What  are  the  parts  ? 


$#  INXKLLECTl.AL      ALGEBRA.  §  18.] 

19.  The  sum  of  the  ages  of  a  man  and  his  wife,  is 
forty-two,  and  twice  the  uian's  age  is  six  less  than 
three  times  his  wife's.     How  old  is  each  1 

20.  Divide  seventeen  into  two  such  parts,  that 
twice  one  part  shall  be  eight  less  than  five  times  the 
other.     What  are  the  numbers  1 

21.  A  farm,  containing  twenty-six  acres,  belongs 
to  two  men.  Tliree  times  A's  part  is  six  acres  less 
than  four  times  B'§  part.     How  many  acres  has  each  1 

22.  Divide  twenty-five  into  two  such  parts,  that 
three  times  one  part  shall  be  three  more  than  five 
times  the  other.     What  are  the  parts  ? 

23.  A  boy,  after  spending  a  part  of  his  money, 
found  he  had  remaining  three  times  as  much  as  he 
had  sj>ent.  He  had  twelve  cents  at  first.  How  much 
did  he  spend  ?  and  how  much  was  left  ? 

24.  A  man  had  thirty-two  sheep.  After  selling  a 
part  of  his  flock,  he  found  the  remainder  was  four  less 
than  twice  the  number  lie  sold.  How  many  did  he 
sell  ?  and  how  many  were  left  ? 

25.  If  IG  —  a;jbe  multiplied  by  2,  what  will  express 
the  product? 

26.  If  10  —  X  be  multiplied  by  7,  what  will  be  the 
product  ? 

27.  In  the  equation  12  —  3x-|-2  =  4  2-,  what  is 
the  value  of  xl 

28.  Reduce  the  equation  26  —  2  z  —  6  =  3  x. 
What  will  be  the  value  of  a;  ? 

29.  Reduce  the  equation  60  —  ox  —  5=:6z. 
What  does  x  represent  ? 

30.  In  the  equation  100  —  3x  —  10  =  7a;  what 
is  the  value  of  x  1 


§    l9  ]  INTELLECTUAL.      ALGEBRA.  97 


SECTIOiN   XIX. 

1.  If  X  be  taken  from  2  x,  the  remainder  will  be  x. 
How  many  more  will  be  left,  if  a:  -^  1  be  subtracted 
from  2  X  1 

Since  not  the  tvhok  of  a;  is  to  be  taken  from  2  :r,  but 
X  less  I  is  to  be  taken  away,  it  is  evident,  that  in 
taking  a\tay  the  whole  of  c,  one  more  is  taken  away 
than  there  should  be. 
This  o}ie,  then,  must  be  put  back,  or  added  to  the  re- 
mainder. 
Therefore,  if  a;  —  1  be  taken  from  2  x,  the  remainder 

must  be  x  and  ene  more,  or  x  -\-  I, 

because,  if  x -{- 1   be  added   to  x  —  1,   the  sum  will 

be  2  X. 

Or, 

it  may  be  expressed  thus  ;  2  z  —  x  -\-l ; 

T   "        uniting  terms,  a;  -f-  1. 

Remark.  —  Therefore,  to  express  subtraction,  change  the 
signs  before  the  quantities  to  be  subtracted,  and-  connect 
them  witii  the  quantities  from  which  thoy  are  to  be  taken 

2.  If  the  expression  x  —  2  be  taken  from  2  x,  what 
will  represent  the  remainder  ? 

3.  If  a;  —  5  be  taken  from  2  a:,  how  many  more 
will  be  left  than  there  would  be,  if  the  whole  of  x 
\vere  taken  from  2  a;?  and  what  will  be  the  remain- 
der ? 

4.  If  x— ^7  be  taken  from  2a-,  how  much  larger 
will  the  remainder  be,  than  if  the  whole  of  x  were 
taken  from  2  a;?  and  what  will  be  the  remainder  ? 


98  INTELLECTUAL  ALGEBRA.       [§  19 

5.  If  2x  —  9  be  taken  from  5x,  what  will  express 
the  remainder  ? 

C.    Peter  has  one  cent  less  tlian  John.     If  Peter's 
money  be  subtracted  from  twice  John's,  the  remainder 
will  be  seven  cents.     How  many  cents  has  each  ? 
Let  X  represent  John's  money  ; 
then  X — 1  will  represent  Peter's. 
Therefore,  if  a;  —  1  be  taken  from  twice  r,  the  dif- 
ference will  be  equal  to  7  cents. 
If  X  be  taken  from  2x,  x  will  remain ;  but  if  07ie  less 

be  taken  from  2  x,  one  more  will  remain. 

Therefore,  if  a:  —  I  be  taken  from  2x,  the  difference 

will  be  x-j-  1- 

Then,  by  the  conditions  of  the  question, 

x-f  1  =  7. 

Subtracting   1   from   each   member   of  the   equation, 

gives 

x=:i6  cents,  or  John's  money ; 

X  —  1  =  5  cents,  or  Peter's  money. 

Reviark. —  In  all  the  examples  in  this  section,  let  x  =  the 
greater,  &c. 

7.  Henry  has  four  cents  less  than  Robert,  and  if 
Henry's  money  be  taken  from  twice  Robert's,  the 
difference  will  be  nine  cents.  How  much  money  has 
each  ? 

8.  The  difference  between  two  numbers  is  five ; 
and  if  the  less  number  be  taken  from  twice  the 
greater,  the  remainder  will  be  seventeen.  What  are 
the  numbers  ? 

9.  The  price  of  a  cow  was  five  dollars  less  than 
'be  price  of  an  ox  ;  and  if  the  p-ic?  o*'  t'^"  c^^v  be 


[§   19.  INTELLECTUAL     ALGEBRA.  99 

taken  from  twice  the  price  of  the  ox,  the  retnainder 
will  be  thirty-five  dollars.  What  was  the  price  of 
each  ? 

10.  The  difference  of  two  numbers  is  twenty-five  ; 
and  if  twice  the  lesS  be  taken  from  three  times  the 
greater,  the  remainder  will  be  eighty.  What  are  the 
numbers? 

11.  A  and  B  gain  money  in  trade,  but  A  receives 
Jen  dollars  less  than  B.  If  A's  share  be  subtracted 
from  twice  B's,  the  remainder  will  be  fifty-seven  dol- 
lars.    How  much  money  did  each  receive  ? 

12.  One  number  is  four  less  than  another,  and  if 
twice  the  less  be  subtracted  from  five  times  the 
greater,  the  remainder  will  be  thirty-eight.  What 
are  the  numbers? 

13.  Two  farms  belong  to  A  and  B.  A  has  twenty 
acres  less  than  B.  If  twice  A's  farm  be  taken  from 
three  times  B's  number  of  acres,  the  remainder  will 
be  one  hundred  acres.     How  many  acres  has  each  ? 

14.  One  number  is  seven  less  than  another,  and  if 
three  times  the  less  be  taken  from  four  times  the 
greater,  the  remainder  will  be  six  times  the  difference 
between  the  two  numbers.   ,  What  are  the  numbers  ? 

15.  Anna  is  four  years  younger  than  Mary.  If 
twice  Anna's  age  be  taken  from  five  times  Mary's, 
the  remainder  will  be  thirty-five  years.  What  is  the 
age  of  each  ? 

'  16.  One  number  is  ten  less  than  another.  If  three 
times  the  less  be  taken  from  five  times  the  greater, 
the  remainder  will  be  seven  times  the  difference  of 
the  two  numbers.     What  are  the  numbers? 

17     Eliza  boufrht  a  doll   and  a  book,  giving  thref 


100         INTELLECTUAL  ALGEBKA.       [§  20. 

cent^  less  for  the  doll  than  for  the  book.  If  twice 
the  price  of  her  doll  be  taken  from  four  times  the 
price  of  her  book,  the  remainder  will  be  forty-six 
cents.     Wiiat  was  the  price  of  each  ? 

18.  If  3a;  — 12  be  taken  from  or,  what  will  rep 
resent  the  remainder  ? 

X  1 

19.  If —  be  taken  from  x,  what  will  express 

the  remainder  ? 

4  a;         3 

20.  If •    be   taken   from   2  j,  what   will   be 

3  4  ' 

the  remainder  ? 

21.  Reduce  the  equation  4  x  —  .i;  -{-  9  :=  15.    What 
does  z  represent  ? 

22.  Reduce    the    equation   5x  —  3  x  -|-  1&'  =  28. 
What  number  does  x  represent  ? 

23.  What  number  does  x  represent  in  the  equation 
Sx  —  2x-{-4=l21 

24.  What  number  will  express  the  value  of  x  in 
the  equation  2x  —  x-\-^  =  2^1 

25.  What  is  the  value  of  x  in  the  equation  Sx  —  .t 
+  7  ;r=  25  ? 


SECTION   XX. 

1.    John  and  William  together  have  twelve  apples 
If  John's  share  be  subtracted  from  twice  the  nunibei 
that  both  have,  the  remainder  will  be  four  times  Wil- 
liam's share.     How  many  apples  has  each  ? 
Let  X  represent  William's  share  ; 


§  20.1  INTKLLECTUAL     AI^GEBRA.  101 

then  12  —  x  will  represent  John's  share. 
Twice  the  number  that  both  ha,ve  is  twice  12,  which 

is  24. 
If  from  24,  12- — x  be  taken,  the  remainder  will  be 

four  times  William's  share  ;  th.it  is,  4  .r. 

If  the  whole  of  12  be  taken  from  24,  x  too  many  will 

be  taken  away,  and  %  must  be  added  to  what  is 

left,  to  give  the  true  remainder  ; 

thus,  24  —  12  -\-  X,  will  express  the  difference. 

Therefore,  by  the  conditions  of  the  question, 

12  -|-  a;  =  4  X. 

Subtracting  x  from  each  member,  gives 

12=3  3;. 

Dividing  each  member  by  3,  gives 

4  =::  a;,  or  William's  share  ; 

12  —  X  z=:  8,  or  John's  share. 

2.  If  twelve  be  taken  from  twenty,  the  remainder 
will  be  eight.  What  more  will  remain  if  12  —  x  be 
taken  from  20  ?  and  what  expression  will  represent 
the  remainder  1 

3.  If  9  —  X  be  taken  from  15,  what  expression  will 
represent  the  remainder  ? 

4.  If  10  — X  be  taken  from  10,  what  will  be  left? 
If  10  be  taken  from  10,  nothing  will  remain;  but  if 

10  diminished  by  the  numbn-  which  x  represents,  be 
taken  from  10,  it  is  evident  that  the  number  that  % 
represents  will  be  the  remainder; 

therefore,  x  will  be  the  remainder. 

It    may   be   expressed   thus;    10  —  10-|-x,  which   is 

equal   to  x. 

5.  If  10  — X  be  taken  from  23,  what  will  express, 
the  remainder  ? 


1()2  INTELLECTUAL     ALGEBRA.  [§  20. 

G.  If  12  —  2  X  be  taken  from  17,  what  express!.  <. 
will  represent  the  remainder  ? 

7.  If,  in  the  above  question,  the  value  oi  x  is  4, 
\^hat  number  will  express  the  remainder  ? 

8.  Robert  and  William  together  have  twenty  cents- 
If -twice  William's  money  be  subtracted  from  three 
times  what  both  have,  the  differenxie  will  be  four  and  a 
half  times  Robert's  money.   Hou^  many  cents  has  each  1 

9.  The  sum  of  two  numbers  is  thirty.  If  the 
greater  be  taken  from  twice  the  sum  of  both,  the 
difference  will  be  equal  to  four  times  the  less  number 
What  are  tlie  numbers  ? 

10.  The  joint  wages  of  two  men  for  one  week  are 
eighteen  dollars ;  and  the  sum  that  each  receives  is 
such,  that  if  twice  B's  money  be  subtracted  from 
three  times  the  amount  that  both  receive,  the  remain- 
der will  be  two  dollars  less  than  four  times  the  money 
which  A  receives.  How  many  dollars  does  each  re- 
ceive ? 

11.  Divide  seven  into  two  such  parts,  that  the  dif- 
ference between  the  larger  and  twice  the  sum  of  both 
will  be  one  more  than  three  times  the  smaller  num- 
ber.    What  are  the  numbers '( 

12.  If  Anna's  age  be  added  to  Susan's,  the  sum 
will  be  fourteen  years  ;  but,  if  Anna's  age  be  taken 
from  three  times  the  sum  of  their  ages,  the  remainder 
will  be  eight  times  Susan's  age.     How  old  is  each? 

Let  X  =z  Susan's  age. 

13.  Divide  forty  into  two  such  parts,  that,  if  tho 
larger  be  taken  from  twice  the  sum  of  both,  the 
smaller  shall  be  three  elevenths  of  tho  remainder 
What  are  the  two  parts  1 


§20] 


INTELLECTUAL     ALGEBRA.  103 


14.  Andrew  and  Peter  have  eleven  oranges,  which 
they  wish  to  divide  in  such  a  manner,  that  the  differ- 
ence between  Andrew's  share  and  twice  the  number 
of  oranges  shall  be  one  orange  less  than  four  times 
Peter's  share.     How  many  oranges  has  each  ? 

15.  The  sum  of  two  numbers  is  sixteen.  If  two 
fifths  of  the  greater  be  taken  from  the  sum  of  both, 
the  less  number  will  be  equal  to  one  half  of  the  dif- 
ference.    What  are  the  numbers? 

16.  Frederic  gave  twenty-four  cents  for  a  book  and 
pencil.  If  the  price  of  the  book  be  taken  from  twice 
the  cost  of  both,  the  difference  will  be  equal  to  four 
times  the  price  of  the  pencil.  What  was  the  price 
of  each  ? 

17.  Divide  twenty-eight  mto  .two  such  parts,  that, 
if  one  fourth  of  the  greater  be  taken  from  the  whole 
number,  the  difference  will  be  twice  the  less  number 
What  will  the  parts  be  ? 

18.  A  cow  and  sheep  cost  thirty  dollars.  If  the 
cost  of  the  cow  be  taken  from  twice  the  cost  of  both, 
the  remainder  will  be  seven  tifnes  the  cost  of  the 
sheep.     What  was  the  cost  of  each  ? 

19.  Divide  thirty-two  into  two  such  parts,  that  if 
four  fifths  of  the  greater  be  taken  from  twice  the 
whole  number,  the  remainder  will  be  four  times  the 
less  number.     What  ate  the  parts  ? 

20.  A  man  and  boy  received  thirteen  dollars  for  a 
week's  labor.  If  two  thirds  of  what  the  man  received 
be  taken  from  twice  the  sum  that  both  had,  the  dif- 
ference will  be  five  times  the  money  which  the  boy 
.'"jceived.     How  many  dollars  had  each? 


104  INTELLECTUAT.  ALGEBRA.       [§  21 


;     SECTION   XXI. 

1.  Eliza  and  Anna  have  ten  books;  and  one  hall 
ot  Anna's  number  of  books  is  equal  to  one  third  of 
Eliza's.     How  many  books  has  each? 

Let  X  represent  Eliza's  number  of  books ; 
then  10  —  X  will  represent  Anna's  number. 

One  half  of  10  —  x  will  be  5  —  -  ; 

2 

but  this  must  be  equal  to  one  third  of  Eliza's,  or—  ; 
therefore,  by  the  conditions  of  the  question. 


Adding  —  to  each  member  of  the  equation, 

-  +  -—5. 
3     '     2 

Reducing  the  terms  of  the  first  member  to  the  same 
denomination,. 

Uniting  terms  in  the  first  member, 

5_a-_  ^ 

6    ~     ' 

If  071(1  sixth  of  5  X  is  5,  the  w^ioh  of  5  x  will  be  six 

times  5,  which  is  30. 

If  5  X  =  30,  X  will  be  ovc  ffth  of  30,  which  is  G  ; 

therefore,  x  =  G,  Eliza's  books, 

and  10  —  x  =  4,  Anna's  books. 

Or, 

snicc    •—  rr  5. 
6 


§  21.]  INTELLECTUAi,     ALOEBRA.  105 

multiplying  each  member  by  6, 

5  a;  =  30. 

Dividing  each  member  of  this  last  equation  by  5, 

x  =  6,  as  above. 

2.  The  sum  of  two  numbers  is  five,  and  one  half 
of  the  less  number  is  equal  to  one  third  of  the  greater 
What  are  the  numbers  ? 

3.  What  is  one  half  of  the  expression  10  —  x? 

4.  If  the  expression  10  —  x  be  divided  by  2,  what 
will  represent  the  quotient  1 

5.  What  is  one  fourth  of  tlie  expression  10  —  x1 
6;    If  the  expressiorv- 1 0  —  x  be  divided  by  4,  what 

will  represent  the  quotient  ? 

7.  If  15  —  2x  be  divided  by  3,  what  will  the  quo- 
tient be  1 

8.  John  and  William  have  seven  oranges.  One 
half  of  William's  number  is  equal  to  two  thirds  of 
John's.     How  many  oranges  has  each  ? 

9.  The  sum  of  two  numl)crs  is  fourteen,  and  one 
half  of  the  greater  is  equal  to  two  thirds  of  the  less 
number.     What  are  the  numbers  ? 

10.  A  and  B  gained  ten  dollars,  and  one  half  of 
A's  share  is  equal  to  one  third  of  B's.  How  many 
dollars  has  each  ? 

11.  The  sum  of  two  numbers  is  eighteen,  and  one 
half  of  the  less  number  is  equal  to  one  fourth  of  tlie 
greater.     What  are  the  numbers  ? 

12.  The  sum  of  the  ages  of  Sarah  and  Caroline  is 
twenty-six  years.  One  third  of  Sarah's  age  is  three 
fourths  of  Caroline's.     What  is  the  age  of  each  ? 

13.  The  sum  of  two  numbers  is  thirteen,  and  one 


106  INTELLECTIML     ALGEBRA.  §^1] 

third  of  the  greater  is  just  three  fourths  of  the  less 
number.     What  are  the  numbers  ? 

14.  Daniel  a"nd  Levi  have  twenty-one  dollars.  If 
Levi's  money  be  divided  by  two,  the  quotient  will  be 
equal  to  Daniel's  money  divided  by  five.  How  many, 
dollars  has  each  ? 

15.  The  sum  of  two  numbers  is  sixteen,  and  the 
less  number  divided  by  three  is  equal  to  the  greater 
divided  by  five.     What  are  the  numbers  ? 

IG.  Divide  twenty-two  dollars  between  A  and  B, 
so  that  if  one  dollar  be  taken  from  three  fourths  of 
B's  share,  and  three  dollars  be  added  to  one  half  of 
A's  money,  the  sums  "shall  be  equal.  How  many  dol- 
lars wrll  each  have  ? 

17.  The  sum  of  two  numbers  is  thirty-three.  If 
one  sixth  of  the  greater  be  subtracted  from  two  thirds 
of  the  less  number,  the  remainder  w'ill  be  seven. 
What  are  the  numbers  1 

18.  Tlie  sum  of  A's  and  B's  money  is  thirty-six 
dollars.  If  five  eighths  of  B's,  less  two  dollars,  be 
taken  from  three  fourths  of  A's,  the  difference  will  be 
seven  dollars.     How  many  dollars  has  each? 

19.  Reduce  the  equation  20 — ^-|~3  =  1''' 

What  number  will  express  the  value  of  x  ? 

20         2  X  1 

20.  In  the   equation f-8 =.5   what 

^  .6  3     '  3 

number  is  represented  hy  x1 


6  22  1  INTELLECTUAL     ALGEBRA.  I0l.7 


SECTION   XXII. 

1.  Anna  paid  eight  cents  for  a  book  and  a  lead- 
pencil,  and  the  pencil  cost  two  cents  less  than  the 
book.     What  did  each  cost  1 

Let  X  represent  the  cost  of  the  book, 

and  let  ?/  represent  the  cost  of  the  pencil ; 

then  x-\-7/  must  express  the  cost  of  both, 

and  z  —  7/  must  be  the  difference  between  the  cos  .  of 

the  pencil  and  book  ; 

therefore,  x-\-i/  =  8,  and  x  —  «/  =  2. 

Adding  x-\-i/  =z8  toz  —  y  =  2,  gives 

x-\-7j-\-x-,j  =  8-\-2. 

Uniting  terms  in  each  member,  gives 

2x=  10. 

Dividing  each  member  by  2,  gives 

X  =1  5  cents,  the  cost  of  the  book. 

Putting  5,  the  value  of  x,  in  the  place  of  x,  in  the 

equation  x-\-i/  =3,  gives 

Taking  5  from  each  member  of  this  last,  gives 

y  r=  3  cents,  the  cost  of  the  pencil. 

Or, 

(1.)  By  a  condition  of  the  question,    .   .   .  x-j-y  rr:^^ 

(2.)  By  another  condl.lcn, x  —  ^  =  2. 

(3.)  Adding  2d  to  1st, 2a;=:zl0 

(4.)  Dividing  3d  by  2, a;  :^  5,  as  above, 

(5.)  Substituting  5,  the  value  of  )  -    ■       q 

X,  for  X  in  the  1st, f 

(6.)  Taking  5  from  each  member  )  o  u 

^    '             *=  >  .  M  =  3,  as  above, 

of  5th )    -^ 


1-08  INTELLECTUAL     ALGEBRA.  [§  22 

2.  The  sum  of  two  numbers  is  eleven,  and  their 
difference  is  three.     What  are  the  numbers? 

Let  X  represent  the  greater  number, 

and  let  y  represent  the  less  number ; 

then  i.-\-y  will  express  their  sum, 

and  X  —  y  will  express  their  difference. 

(1.)  Then,  by  a  condition  of  the  question,  x-\-  y  ^=.  11 

(2.y  And,  by  another  condition, x  —  y  ^=  3. 

(3.)  Adding  2d  to  1st, 2i=14. 

(4.)  Dividing  3d  by  2,    .  .  a;  =:  7,  the  greater  number. 

(5.)   Subtracting  x  from  each  mem-  )  i, 

^     '                      °  /  .   .  y  z=:  11  —  X. 

ber  of  1st, ' 

(6.)  Putting  7,  the  value  of  x,  for  a-,  )  , ,       -, 

in  5th, i        ^ 

Then  3^  =  4,  the  smaller  number. 

3.  The  sum  of  the  ages  of  two  boys  is  twelve 
years,  and  the  difference  of  their  ages  is  six  years. 
What  are  their  ages  1 

4.  If  x-\-y  be  added  to  x  —  y,  what  expression 
will  represent  their  sum  ? 

5.  If  3x-|-3/  be  added  to  Ax  —  y,  what  will  ex- 
press the  sum  ? 

6.  If  2x  —  3y  be  added  to  3  x  +  3  y ,  what  will 
represent  their  sum  1 

7.  If  the  equations  —  yz^S,  be  added  to  the 
equation  x-\- y  =z7,  what  equation  will  express  the 
sum  of  the  two  ?  and  what  are  the  respective  values 
of  x  and  y  ? 

8.  If  the  equation  x-\-  y^^lO,  be  added  to  the 
equation  3x  —  ?/ =  IG,  what  new  equation  will  result 
from  the  addition  ?  What  are  the  respective  values 
of  X  and  y  ? 


§22] 


INTELLECTUAL     ALGEBRA.  109 


9.  If  2x  —  1/=:  11  be  added  to  the  equatiou  4x-(- 
y  =  31,  what  will  be  the  equation  expressing  their 
sum  1  and  what  numbers  do  x  and  y  respectively  rep- 
resent ? 

10.  If  4x  —  i/  =  27  be  added  to  2x-[-y  =  29, 
what  equation  will  express  the  result  1  and  what  will 
be  the  respective  values  of  x  and  y  ? 

11.  U3x  —  2i/z=zl8  be  added  to  2  X -f  2  y  =  22, 
what  will  be  the  sum  of  the  two  equations?  What 
will  be  the  value  of  x,   and  what  the  value  of  y  ? 

12.  If  X  -|-  3  y  rr  28  be  added  to  the  equation  3  x 
—  3y::3l2,  what  equation  will  express  the  result? 
What  is  the  value  of  x  and  y,  eacli  1 

13.  If  the  equation  y  r::  y  be  added  to  the  equa- 
tion X  —  y  zz:  2,  that  is,  ify  be  added  to  each  member 
of  this  last  equation,  what  will  the  equation  become  1 

14.  Charles  bought  five  peaches  and  two  pears  for 
seventeen  cents,  and  found  that  two  pears  cost  four 
cents  less  than  two  peaches.  What  did  one  of  each 
cost? 

15.  There  are  two  numbers,  such  that,  if  three 
times  the  greater  be  added  to  three  times  the  less,  the 
sum  will  be  twenty-one ;  and  if  three  times  the  less 
be  taken  from  five  times  the  greater,  the  remainder 
will  be  nineteen.     What  are  the  numbers  ? 

IG.  If  three  times  Anna's  age  be  added  to  three 
limes  Mary's  age,  the  sum  will  be  thirty-three  years ; 
and  three  times  Mary's  age  is  thirty-seven  years  less 
than  seven  times  Anna's.  What  are  their  respective 
ages  ? 

17.    Four  times  the  sum  of  two  numbers  is  twenty 
eight,  and  the  difference  between  six  times  the  greater 


no  IXTRLLECTUAL     ALGEBRA.  [^  22. 

and  four  times  the  less  is  twelve.     What  are  the  num- 
bers ? 

18.  A  market-man  sold  three  melons  and  four 
peaches  for  thirty-eight  cents,  and,  at  the  same  prices, 
five  melons  would  sell  for  forty-two  cents  more  than 
he  received  for  the  four  peaches.  What  did  he  re- 
ceive for  one  of  each  ? 

19.  The  difference  between  two  numbers  is  seven, 
and  four  times  the  greater  added  to  the  less  is  forty- 
tliree.     What  are  the  numbers  ? 

20.  A  farmer  bought  three  sheep  and  a  cow  for 
twenty-six  dollars.  At  the  same  rate,  a  cow  would 
cost  four  dollars  less  than  twelve  sheep.  What  did 
he  pay  for  the  cow,   and  what  for  a  sheep  ?   "^ 

21.  Twice  the  smaller  of  two  numbers,  taken  from 
three  times  the  larger,  leaves  only  fourteen;  and,  if 
the  larger  be  added  to  twice  the  smaller,  the  sum  will 
be  eighteen.     What  are  the  numbers  ? 

22.  If  three  fourths  of  John's  age  be  taken  from 
William's  age,  the  difference  will  be  six  years ;  but  if 
five  times  William's  age  be  added  to  three  fourths  of 
John's,  the  sum  will  be  sixty-six  years.  What  is  the 
age  of  each  1 

23.  One  number  is  two  more  than  twice  another, 
and  the  sum  of  tour  times  the  larger  and  twice  the 
smaller  is  forty-eight.     What  are  the  numbers  ? 

24.  The  sum  of  one  seventh  of  Anna's  money  and 
one  third  of  George's  is  six  cents,  and  if  one  third  of 
George's  be  taken  from  four  sevenths  of  Anna's,  the 
remainder  will  be  nine  events.  How  many  cents  had 
each  1 

25.  Divide  twenty  into  two  such  parts,  that  tLe  dif- 


INTELLECTUAL     ALGEBRA.  ill 

ference  between  the  smaller  and  three  times  the  larger 
will  be  twenty-four.     What  are  the  parts  ? 

20.    The  sum  of  two  numbers  is  fourteen,  and  the 

diflerence  between   four  times  the  greater  and  twice 

the  less  is  twenty-six.     What  are  the  numbers  ? 

Let  X  =  the  greater  number, 

and  ?/  =:  the  smaller  number. 

(1.)  By  one  condition  of  the  question,    .    x  -j-  ?/  =:  14. 

(2.)  By  another  condition, 4x  —  2y=r26. 

(3.)  Adding  1st  to  itself, 22;-|-2y  =  28. 

(4.)  Adding  3d  and  2d, 6  a;  =  54. 

(5.)  Dividing  each  member  of  )  x  =  9,    the    greater 

4th  by  6, S  number. 

(6.)   Subtractincr   %   from  each  )  ,  . 

^    '                     °  >  .  .  .  .    y  =  14  —  X. 

member  of  1st, ' 

(7.)  Substituting  9,  the  value  |?/  =  14  —  9,  or  5,  the 

of  a-,  in  6th, )       smaller  number. 

27.  If  x-f-7/rr9  be  added  to  x-\-7/  =  9,  what 
equation  will  be  formed  ? 

28.  If3x  — 2y  =  21  be  added  to5x  +  2  7/  =  67, 
what  equation  will  express  the  sum  1  and  what  num- 
bers do  X  and  i/  represent? 

4x32/  .         2  X 

29.  If =  2  be  added  to  the  equation   — 

3y  . 

-\ =r  10,  what   will   express  the   sum?    and   what 

numbers  do  x  and  y  represent? 

30.  Add  the  equation  ^  -|-  -|-  rr  18  to  the  equation 

u  .T        y 

— =  7.    What  equation  will  express  the  sum? 

and  what  numbers   are  represented  by  x   and  y  re- 
spectively ? 


11-3  INTELLECTUAL     aLGEBKA.  FA  23. 

31.  If  the  equation  x-\-i/=zG  be  added  to  itself, 
what  equation  will  express  the  sum  .'  that  is,  if  z  -f- 
y  =  G  be  multiplied  by  2,  what  will  be  the  product  ? 

32.  If  z -["S' =  ^  ^6  subtracted  from2x-|-2y  = 
12,  what  equation  will  express  the  remainder  ?  that  is, 
if  the  ecjuation  2  x -\- 2  i/=z  12  be  divided  by  2,  what 
jsquation  will  express  the  quotient  ? 

33.  WJiat  is  one  third  of  the  equation  x-\-y  =^61 

34.  If  the  equation  x-\-i/=zij  be  divided  by  2, 
what  equation  will  express  the  quotient? 

35.  If  the  equation  x  —  y  =  2  be  added  to  itself, 
what  equation  will  represent  the  sum? 

36.  If  the  equation  x  —  y  =-2  be  multiplied  by  2, 
what  new  equation  will  be  produced  ? 

37.  If  X  —  y=^2  be  multiplied  by  4,  what  equa- 
tion will  express  the  product  1 

38.  If  X  —  i/=:2he  subtracted  from  the  equation 
2x- — 2  y  zir  4,  what  equation  will  represent  the 
remainder  ? 

39.  If  the  equation  2x  —  2  ?/  =  4  be  divided  by  2, 
what  equation  will  represent  the  quotient  ? 

40.  What  is  one  half  of  the  equation  2x  —  2?/ 
=  4? 


SECTION   XXIII. 

I.  A  FARMER  sold  to  oiie  man  two  sheep  and  three 
lambs  for  seven  dollars,  and  to  anotlie*  at  the  saiiu; 
rate,  one  lamb  and  two  sheep  for  five  Hollars.  What 
did  he  receive  for  one  of  e.ich? 


<j23.] 


INTELLECTUAL     ALGEJBKA.  1  13 


Let  3;=  the  price  of  a  sheep, 

and  y  =  the  price  of  a  lamb  ; 

then  2  X  =  the  value  of  two  sheep, 

and  3  ?/  =  the  value  of  three  lambs. 

But  two  sheep  and  three  lambs  were  sold  for  seven 

dollars ; 

therefore,  by  the  first  condition  of  the  question, 

the  first  equation  will  be, 

2  a; +  3?/ =  7. 

Also  one    lamb    and   two   sheep   were    sold   for   five 

dollars ; 

therefore,  by  the  second  condition  of  the  question,  the 

second  equation  will  be 

2x-|-y  =  5. 

Now,   if  2  X -\- 7/  be  taken  from  2  x-\-  3 y,2  1/  only 

will  remain, 
and  2  y  will  be  equal  to  the  difference  between  5  and  7  ; 

therefore,  2?/  =:  2,  and  j^  r=:  1. 

Since  y  =  the  price  of  a  lamb,  he  sold  a  lamb  for  one 

dollar. 

In  the  equation 

2  X  +  y  =  5, 

substituting  l,the  value  of  1/,  instead  of  ?/,        , 

2  X  -f  1  =  5. 

If  2  X  and  1  more  =  5,  2  x  alone  :=  4, 

and  X  ^=2. 

But  x=  the  price  of  a   sheep;  therefore  he  sold  a 

sheep  fi)r  two  dollars. 

Or, 

(I.)  By  a  condition  of  the  question,  .    2x-|-3y:=7 

(2.)  And,   by  another  condition,     .   .   .   2x-{-7/=:5. 

(3.)  Subtractnig  2d  from  1st,     2v~2 


H4  INTELLECTUAL     ALGEBRA  [§23. 

(4.)  Dividing  3d'by  2, y  =  l- 

(5.)  Subtracting  y  from  2d, 2  x  :r=  5  —  y. 

(6.)   Substiiutincr  1,  the  value  of )    ^  -       ,  . 

^    '            .          °    '  >   2x  =  5  —  1,  or  4. 

y,  in  the  5th,    ) 

(7.)  Dividing  6th  by  2, a:  =  2,  as  above. 

2.  The  sum  of  two  numbers  is  twelve  ;  and  if 
twice  the  greater  be  added  to  the  less,  the  sum  will 
be  nineteen.     What  are  the  numbers  ? 

Let  z  =.  the  greater  number, 

and  y  =z  the  smaller  number. 

(1.)  By  a  condition  of  the  question,  .    2x-|-y=:  19 

(2.)  By  another  condition, x-\-i/=l2 

(3.)  Subtracting  2d  from  1st,  (     x=  ', 

(  the  greater  number. 

(4.)  Subtracting  X  from  2d, iV=12 — x- 

(5.)   Substituting  7,  the  value  of  i  ?/  =  12  —  7,  or  5, 

■  X,  in  4th, )       the  smaller. 

* 

3.  If  the  expression  x-\-  y  he  taken  from  3  x  -j-  y, 
what  win  represent  the  remainder? 

4.  If  2x-\-2y  be  taken  from  2x-\-5y,  what  will 
be  the  remainder  1 

5.  If  2  x  -{-  3  y  be  taken  from  7  z  -j-  3  y,  what  will 
remain  ? 

6.  If  r  -|-  4  7/  be  taken  from  3  x  -J-  4  y,  what  vvilJ 
express  the  difference  ? 

7.  If  X  -{-  2y  be  taken  from  x  -\-9  y,  what  will  be 
the  difference  ? 

8.  If  the  equation  2x-}-v/ =  II  be  taken  from  the 
equation  2  x -|- 9y  =:  35,  what  equation  will  express 
the  difference  ?  and  what  are  the  values  of  x  and  y 
respectively  ? 


§  23.] 


INTELLECTUAL     ALGEBRA.  115 


9.  If  the  equation  x-\-2  y  =  V3  be  subtracted 
from  the  equation  5  z-{-  2y  =  25,  what  equation  wil/ 
express  the  remainder  ?  and  what  arer  the  respective 
values  of  x  and  5  ? 

10.  George  paid  sixteen  cents  for  two  pens  and 
four  pencils.  Charles,  buying  at  the  same  price,  paid 
ten  cents  for  two  pens  and  two  pencils.  What  was 
paid  for  a  pen,  and  what  for  a  pencil  ? 

11.  The  sum  of  two  numbers  is  thirteen ;  and 
three  times  the  greater  added  to  the  less  is  twenty- 
seven.     What  are  the  numbers? 

12.  A  man  sold  five  lemons  and  two  oranges  for 
twenty-two  cents;  and  again  he  sold,  at  the  same 
rate,  two  oranges  and  three  lemons  for  eighteen 
cents-.     What  did  he  receive  for  one  of  each? 

13.  The  sum  of  two  numbers  is  seven.  If  five 
times  the  less  be  added  to  the  greater,  the  sum  will 
be  fifteen.     What  are  the  numbers? 

14.  A  boy  paid  thirty-nine  cents  for  tliree  lead 
pencils  and  five  writing-books,  and  he  afterwards 
purchased,  at  the  same  rate,  three  pencils  and  three 
writing-books  for  twenty-seven  cents.  What  did  he 
pay  for  one  of  each? 

15.  Find  two  such  numbers,  that  the  sum  of  twice 
the  greater  added  to  six  times  the  less,  will  be  thirty- 
six,  and  the  sum  of  three  times  the  less  added  to 
twice  the  greater  will  be  twenty-four.  What  are  the 
numbers  ? 

16.  Eliza  bought  two  peaches  and  seven  pears  for 
twenty' cents  ;  and  again,  at  the  same  rate,  she  bought 
three  pears  and  two  peaches  for  twelve  cents.  What 
did  she  pay  for  one  of  each  ? 


I 16  INTELLECTUAL     ALGEBRA.  T^  2d 

17.  There  are  Iwo  numbers  such  that,  if  five  times 
the  greater  be  added  to  three  times  the  less,  the  sum 
will  be  forty-four;  and  if  t^iree  times  the  less  be 
added  to  the  greater,  the  sum  will  be  sixteen.  What 
are  the  numbers  ? 

18.  Twice  George's  age  added  to  five  times  Lucy's 
is  forty-three  years,  and  twice  Lucy's  added  to  twice 
George's  is  twenty-two  years.     How  old  is  each  ? 

19.  Find  two  numbers,  such  that  the  sum  of  three 
times  the  greater  added  to  eight  times  the  less  will  be 
forty-seven,  and  the  sum  of  twice  the  less  added  to 
three  times  the  greater  will  be  twenty-three.  What 
are  the  numbers  ? 

20.  A  man  bought  a  coW  and  ten  sheep  for  forty 
dollars.  He  then  sold,  at  the  same  rate,  seven  sheep 
and  a  cow  for  thirty-four  dollars.  What  was  the 
price  of  one  of  each  ? 

21.  John  said  to  Henry,  "If  one  half  of  my  money 
be  added  to  two  thirds  of  yours,  the  sum  will  be  ten 
dollars."  Henry  replied,  "  If  one  third  of  my  money 
be  added  to  one  half  of  yours,  the  sum  will  be  seven 
dollars."     How  many  dollars  had  each? 

22.  Tliere  are  two  numbers  such,  that  if  three 
fourths  of  the  greater  be  added  to  two  thirds  of  the 
less,  the  sum  will  be  twenty-five ;  but  if  two  thirds  of 
the  less  be  added  to  one  fourth  of  the  greater,  the 
sum  will  be  only  fifteen.     What  are  the  numbers  ? 

23.  If  the  equation  3  .t -|- 3  ?/ :=  21  be  subtracted 
from  the  equation  3  a;  -|-  5  y  =:  27,  what  equation  will 
represent  the  remainder?  and  what  will  be  the  re- 
spective values  of  x  and  y  ? 

24.  Ux-\-2i/^\6  be  taken   from  9  x -j-2  i/ = 


§  23.]  INTELLECTUAL     ALGEBKA.  117 

32,  what  equation  will   express  the  difference?    and 
what  will  be  the  respective  values  of  %  and  yl 

25.  If  the  expression  —  -|-  -^  be  subtracted  from 

the  expression  —  -| -,  what  will  represent  the  dif- 
ference ? 

26.  If  from  the  equation  "-^  -\ =  14,  the  equa- 

2  a:         2  7/ 

tion  — ■  -| =  8  be  taken,  what  equation  will  result 

from  the  subtraction  ?  and  what  will  be  the  respective 
values  of  x  and  y  ? 

27.  Subtract    -i^-f-^^rll    from  —  +  ^  z=  23. 

8      '      4  8      '      4 

What  will  be  the  remainder  ?   and  what  numbers  do 
X  and  y  respectively  represent  % 

28.  Reduce  the  equations  3 1  •  |-  7  ?/  r=  29,  and  x  -j- 
2y  =  9,  the  respective  values  of  a;  and  y  being  the 
same  in  each  equation.  What  number  does  x  and  y 
each  represent  ? 

(1.) x  +  2y  =  9. 

(2.) 32;  +  7y  =  29. 

(3.)  Multiplying  1st  by  3, 3  2;-i-6y  =  27. 

(4.)  Subtracting  3d  from  2d, «/  =  2. 

(5.)  Taking   2y    from    each  ^ 3._9_.o„ 

member  of  the  1st,  .  .  S 
(6.)  Substituting  2,  the  value  )        (\      a  k 

of  y,  in  5th, J 

Therefore,  the  number  represented  by  y  is  2, 
and  the  number  represented  by  x  is  5. 

29.  If  X   and   y  respectively  represent   the   same 


lis  INTELLECTUAL      ALGEBRA. 


[§24. 


numbers  in  the  equations  o  r -)- 2  j/ =r  26,  and  oi-f* 
81/  z=i  44,  what  will  be  the  value  of  each  ? 

80.    1£  X -{-  1/  =z  7  he  taken   from  2  x  -\-  4  y  z=  20, 
what  will  express  the  difference  ? 

31.  If  2  a;  -f-  4  y  =  20  be  divided  by  2,  what  will 
be  the  quotient  ? 

32.  What  is  one  half  of  4  x  -f-  6  y  =  34  ? 

33.  Divide  3^4-9^  =  39  by  3.     What  will  be. 
the  quotient  1 

34.  What  is  one  third  of  6  .t  +  3  1/  =  33  ? 

35.  What  is  two  thirds  of  6  a:  +  37/  —  33  ? 

36.  What  is  three  fourths  of  8  x  +  12  5/  =  28  ? 

37.  Divide  x -\- 1/ z=  6  by  2.     What  will  be   the 
quotient  ? 

38.  What  is  one  third  of  2  z  +  y  :=  12  ? 

39.  What  is  two  thirds  of  2  x  +  y  =  12  ? 

40.  W^hat  is  three  fourths  of  2  z  -|-  y  =:  12  '' 


SECTION   XXTV 

1.    A  BOY  bought  five  oranges  and  two  lemons  for 

twenty-six  cents,  and  a  lemon  cost  one  cent  less  than 

an  orange.     What  was  the  price  of  one  of  each  ? 

Let  X  =z  the  price  of  an  orange, 

and  ?/  ^-  the  price  of  a  lemon. 

(1.)  By  a  condition  of  the  question,  .  .  .  .x  —  t/ =  I 

(2.)  By  another  condition, 5  x -f- 2  y  =:  26. 

(3.)  Multiplying  1st  by  2, 2z  — 2yr=:2. 

'4.)  Adding  2d  and  3d, 72:=:28 


A  24.  ]  INTELLECTUAL     ALGEBRA.  I  [V 

(5.)   Dividing  4th  by  7,    .  .  x=:  i,  price  of  an  orange. 
(6.)   Subtracting  5  x  from  each  )  ^^     ^^^ ^ 

member  of  2d,   .  .  .  .   ) 
(7.)  Substituting  4,  the  value  )^    ^         2C)  —  20  =  G. 

of  X,  for  X,  in  6th,    .  .   ) 
(8.)  Dividing  7th  by  2,    .  .   .  y  zz:  3,  price  of  a  lemon. 
An  orange  cost  4  cents,  and  a  lemon  3  cents. 

2.  The  difference  of  two  numbers  is  three,  and  the 
sum  of  four  times  the  greater  added  to  three  times  the 
less  is  twenty-six.     What  are  the  numbers  ? 

Let  X  =  the  greater  number, 

and  y  :=  the  smaller  number. 

(1.)   By  a  condition  of  the  question,  .   .   .   .x  —  i/=z3. 

(2.)  By  another  condition, 4x-|-3y  =  26. 

(3.)   Multiplying  1st  by  3, Sx  —  3y  =  9. 

(4.)   Adding  2d  to  3d, 72;  =  35. 

(5.)  Dividing  4th  by  7, a;  =  5,  the  greater. 

(6.)  Subtracting  4  X  from  2d,  .  .  .  .   3y  =  26  —  4  x. 

(7.)  Substituting  5,  the  value  of  ^  .^     (,p j,^ ^ 

X,  for  x,  in  6th, ' 

(8.)  Dividing  7th  by  3, y  =  2,  the  smaller. 

The  greater  number  is  5,  and  the  smaller  2. 

3.  If  twice  the  expression  x  —  y  be  added  to  the 
expression  3  x  -|-  2  y,  what  will  represent  the  sum  ? 

4.  If  five  times  the  expression  2  x  —  y  be  added  to 
3  X  -|-  5  y,  what  will  express  the  sum  ? 

5.  If  twice  the  equation  x  —  y=I  be  added  to 
the  equation  3  x  -|-  2  y  =  8,  what  equation  will  rep- 
resent the  sum  ?  and  what  will  be  the  respective 
values  of  x  and  y  ?    ^  —  y  ^^       /y  ~    /, 

6.  If   three   times    the    equation  2x  —  y  =  5   be 


IW  INTELLECTUAL     ALGEBRA.  T^  24. 

added  to  tJie  equation  4  i  -|-  3  ?/=^25,  Avhat  new  cqua 
tion  will  result  from  such  addition  ?  and  what  num 
bers  do  x  and  1/  re^ipectively  represent  ? 

7.  Multipl-y  the  equation  2  x  —  2yr=4  by  2,  and 
add  the  product  to  the  equation  3  a* -|- 4  y  rr  20. 
What  will  represent  the  result  ?  and  what-  are  the 
respective  values  of  x  and  1/  1     ^ 

8.  If  four  times  the  equation  x  —  ^  =  o  be  added 
to  2  a:  -{-  4  ?/  =  22,  what  equation  will  be  formed  ? 
What  will   be  tl)e  value  of  x,   and  what  of?/? 

9.  If  three  times  the  equation  2  x  —  2  ?/  =  4  be 
added  to  4  x -\- G  1/ =  dS,  what  equation  will  result? 
and  what  will  be  the  value  of  each  of  the  unknown 
quantities  x  and  .y  ?         . 

10.  A  farmer  sold  a  cow  and  a  caii*  for  twenty-five 
dollars.  At  the  same  rate,  three  calves  would  be  sold 
for  five  dollars  less  than  what  he  obtained  for  the  cow. 
IIow  many  dollars  did  he  get  for  each  ? 

11.  There  are  two  numbers,  such  that,  if  twice  the 
greater  be  taken  from  three  times  the  less,  the  re- 
mainder will  be  two,  and  if  four  ^iines  the  less  be 
added  to  the  greater,  the  sum  will  be  twenty-one. 
What  are  the  numbers  ? 

12.  If  three  times  Eliza's  age  be  added  to  four 
times  Clara's,  the  sum  will  be  thirty-eight,  and  if 
Clara's  be  taken  from  twice  Eliza's,  the  remainder 
will  be  seven.     W^hat  is  the  age  of  each? 

13.  Find  tw*^.  numbers,  such  that  the  sum  of  three 
times  the  greater  and  twice  the  less  shall  be  twenty- 
one,  and,  i^  iight  times  the  less  be  subtracted  from 
nine  times  tVit;  greater,  the  remainder  will  be  twenty- 
one.     What  arc  the  numbers  ? 


i;24.j 


INTELLECTUAL     ALGEBRA.  12  J 


14.  If  three  times  George's  books  be  added  to 
Mary's,  the  sum  will  be  twenty-three;  but  if  five  times 
Mary's  be  taken  from  five  times  George's,  the  remain- 
der will  be  twenty-five.     liow  many  books  has  each? 

15.  Divide  seventeen  into  two  such  parts,  that  the 
difference  between  three  times  the  smaller  and  five 
times  the  larger  shall  be  forty-five.  What  are  the 
parts? 

16.  Two  men  in  partnership  divide  their  gain,  sc 
that  the  sum  of  twice  A's  share,  added  to  B's  share, 
will  be  twenty-seven  dollars ;  and  if  three  times  B's 
money  be  taken  from  four  times  A's,  nineteen  dollars 
will  be  left.     How  many  dollars  will  each  have? 

17.  There  are  two  numbers  whose  difference  is 
four,  and  if  three  times  the  greater  be  added  to  three 
times  the  less,  the  sum  will  be  thirty.  What  are  the 
numbers  ? 

18.  A  melon  and  an  orange  together  cost  twenty 
cents,  and  one  fourth  of  an  orange  cost  three  cents 
less  than  one  fourth  of  a  melon.    V/hat  did  each  cost? 

Let  X  =:  the  cost  of  a  melon, 

and  1/  =  the  cost  of  an  orange. 

X        y 
(1.)   Bv  a  condition  of  the  question,    .  .   . =  3. 

\      I        •  4  4 

(2.)   By  another  condition, a: -]-?/  =  20. 

(3.)  Multiplying  1st  by  4, :i^_^:=1.2. 

(4.)  Reducing  fractions  in  3d, x — y  r=  12. 

(5.)   Adding  2d  and  4th, 2a;=r:32. 

(6.)  Dividing  5th  by  2, a;rr:16. 

(7.)  Taking  x  from  each  member  ^  ^p. 

in  the  2d, ^   ■  ■  •  V 


122  INTELLECTUAL  ALGEBRA.        [§  24. 

(8  )  Substituting  IG,  the  value  off  r.->       i^. 

'  ^         f     ,  -        >  1/ =  20  —  Jb  =  4. 

X,  for  X,  in  the  7th,  .  .  .  .  ; 

A  melon  cost  IG  cents,  and  an  orange  4  cents. 

19.  There  are  two  numbers,  such  that,  if  one  fifth 
of  the  smaller  be  taken  from  one  fifth  of  the  larger, 
the  remainder  will  be  one,  and  if  three  times  the  larger 
be  added  to  the  smaller,  the  sum  will  be  thirty-five. 
What  are  the  numbers  ? 

20.  If  due  half  of  the  price  of  a  horse  be  adde<l 
to  one  fourth  of  the  price  of  a  cow,  the  sum  will  be 
forty-six  dollars ;  but  if  one  eighth  of  the  price  of  the 
cow  be  taken  from  one  eighth  of  the  price  of  the 
horse,  the  remainder  will  be  seven  dollars.  What  is 
the  price  of  each  ? 

21.  If  one  sixth  of  a  less  number  be  taken  from 
one  third  of  a  larger,  the  remainder  will  be  three, 
and  if  two  thirds  of  the  larger  be  added  to  one  half 
of  the  smaller,  the  sum  will  be  sixteen.  What  are 
the  numbers  1 

22.  A  man  said,  that  if  one  half  the  price  of  his 
saddle  were  taken  from  one  fiTih  of  the  price  of  his 
horse,  the  difference  would  be  fifteen  dollars ;  but  one 
tenth  of  the  price  of  his  horse  and  one  tenth  of  the 
price  of  his  saddle  together  would  he  eleven  dollars. 
What  was  the  price  of  each  ? 

23.  If  one  half  of  the  greater  number  be  added  to 
the  whole  of  the  less,  the  sum  will  be  seven  ;  but  if 
one  half  of  the  less  be  taken  from  the  whole  of  the 
greater,  the  remainder  will  be  four.  What  are  the 
numbers  1 

24.  If  twice  2  x  —  .'/  =  ''   be   added  to  o  x  -f-  2  ^ 


§24.]       INTELLECTUAL  ALGEBRA.  123 

:rr21,  wliat  equation  will  express  tlie  result?  and 
what  are  the  respective  values  of  x  and  y  1 

X       y 

25.  If    four  times  the  equation    ~ :=:  1    be 

added  to  x -\-  y  ^=.  20,  what  equation  will  represent 
the  sum?  and  what  are  the  rfespective  values  of  x 
and  y  1 

y 

26.  If    three    times   the  equation   x =::  7  be 

added  to  4  x -\- y  =  42,  what  equation  will  express 
the  sum  ?     What  do  x  and  y  represent  ? 

y 

27.  If    four    times    the     equation    x ^  =  9  be 

8 

y 
:.dded  to  the  equation  5  a:  -j-  —  =  54,  what  equation 

will  be  formed  ?  and  what  will  be  the  respective 
values  of  x  and  y  ? 

2  .T        y 

28.  If  three  times  the  equation  -^ =10   be 

^  3         G 

X       y 
added  to  twice  the  equation 1 m  12,  what  equa- 
tion  will   express   the   sum  ?    and  what    will    be    the 

respective  valu6s  of  x  and  y  ? 

y 

29.  If    four    times    the    equation    x =  4   be 

y 
added   to  three  times  the  equation  2  x -\ rrllJ-, 

what  equation  will  result?  and  what  will  be  the  re 
ppective  values  of  x  and  y  ? 


l'2'l  JNTELLECTUAL     ALGEBRA.  .[§  2b. 


SECTION    XXV. 

1.  A  FARMER  sold  five  barrels  of  pears  and  four 
barrels  of  apples  for  twenty-three  dollars.  He  after- 
wards sold,  at  the  same  rate,  two  barrels  of  each  for 
ten  dollars.     AVhat  was  the  price  of  a  barrel  of  each  1 

Let  x=rthe  price  of  a  barrel  of  pears, 
and  y  :=z  the  price  of  a  barrel  of  apples. 

(1.)  By  one  condition  of  the  question,  2  a; -f- 2?/ :=:  10 

(2.)  By  another  condition, 5  x -\- i  ?/ z=z  2^. 

(3.)  Multiplying  1st  by  2, 4  a: -j- 4  ^r  =:  20. 

(4.)  Subtracting  3d  from  2d, x  =  S. 

(5.)  Taking  2x  from  each  mem-  \      ey     in o 

ber  of  1st, y 

(().)  Dividing  5th  by  2,   ". ]/ =^  o  —  x 

(7.)  Substitutinor  3,  the  value  of  )  r       .->       f^ 

^    '            .          <='  y  .  1/  =z  5  —  3=:  2. 

X,  in  the  6th, ) 

A  barrel  of  pears  cost  3  dollars,  and  a  barrel  of  apples 

cost  2  dollars. 

2.  There  are  two  numbers,  such  that  three  times 
the  greater  added  to  the  less  is  fourteen,  and  three 
times  the  less  added  to  the  greater  is  ton.  What  are 
the  numbers  ? 

3.  If  twice  the  exprcasion  x  -\-  y  be  subtracted 
from  the  expression  2x-{-3y,  what  will  represent  the 
remainder  ? 

4.  If  2  X -]-//,  multiplied  by  4,  be  subtracted  from 
8  x -]- G  ?/,  what  will  represent  the  remainder? 

o.  If  twice  the  equation  3x-f-2y  =  8  be  sub- 
tracted from  the  equation  Tx  -j-4y  =  18,  what  equa- 


^25.]  INTELLECTUAL     ALGEBRA.  125 

lion  will    rf.'prcsent   the   remainder  '>.      What   number 
does  X  and  ?/  eacli  represent  ? 

6.  If  from  three  times  the  equation  2,  x -\- i/ =1  10 
the  equation  4  x -\- 2 y  =z  22  be  taken,  what  equation 
will  express  the  result  1  What  will  be  the  respective 
values  of  x  aiid  y  1 

7.  Multiply  the  equation  x-\-2  i/  =  7  by  four,  anu 
from  the  product  subtract  the  equation  4a;-[-4  7/r= 
IG.  What  equation  will  express  the  result  ?  What 
will  be  the  values  of  x  and  y,  each  ? 

8.  When  twice,  Mary's  age  is  added  to  three  times 
Jane's,  the  sum  is  nineteen ;  and  when  three  times 
Mary's  age  is  added  to  Jane's,  the  sum  is  eighteen. 
What  is  the  age  of  each  ? 

9.  When  the  greater  of  two  numbers  is  added  to 
twice  the  less,  the  sum  is  fourteen ;  and  when  the  less 
is  added  to  twice  the  greater,  th.e  sum  is  sixteen. 
What  are  the  numbers  ? 

10.  Martha  bought  three  pencils  and  a  book  for 
nineteen  cents.  Abby  bought,  at  the  same  rate,  a 
pencil  and  two  books  for  twenty-three  cents.  What 
was  the  cost  of  one  of  each  ? 

11.  There  are  two  nundsers,  such  tliat,  if  live  times 
the  greater  be  added  to  four  tinTes  the  less,  the  sum 
will  be  thirty-eight,  and  if  twice  the  less  be  added  to 
the  greater,  the  sum  will  be  ten.  What  are  the  num- 
bers ? 

12.  A  farmer  sold  five  sheep  and  four  Iambs  for 
twenty-three  dollars.  He  afterwards  bought,  at  the 
same  rate,  a  sheep  and  two  lambs  for  seven  dollars 
What  was  the  price  of  one  of  each  ? 

13.  Divide  twenty-five   into  two  such   parts,   thaj 


12(3  INTELLFXTUAL     ALGEBRA.  [§^5 

the  sum  of  three  times  the  less  and  four  times  the 
greater  shall  be  only  five  less  than  four  tiroes  the  suir. 
of  both  parts.     What  are  the  parts  ? 

14.  If  John  gives  you  two  thirds  of  his  apples,  and 
Henry  gives  you  one  half  of  his,  you  will  receive 
twelve ;  but  if  John  gives  you  one  sixth  of  his,  ami 
Henry  gives  you  one  fourth  of  his,  you  will  get  only 
four.     How  many  apples  has  each  ? 

Let  X  =z  John's  number  of  apples, 
and  y  r=  Henry's  number  of  apples. 


(2. 
(3. 

(4. 
(5. 
(G. 

(-• 
(8. 


By  one  condition  of  the  question,     . 1 :=:  4. 

By  another  condition, - — \-  —  :=.VZ. 

Multiplying  1st  by  2, .^  +  1^S 


Subtracting  3d  from  2d —  =  4. 

o  '  3 

Multiplying  4th  by  3, x  =  12 

Taking  —  from  each  member  of  3d,  —  r=  8 

3  2  3 

Putting  12  for  x  in  the  6th,     .  —  =  8  —  —  =  4. 
°  2  3 


Multiplying  7th  by  2, y  =  8 

John  had  12,  and  Henry  had  8  apples. 

15.  The  sum  of  two  thirds  of  the  greater  -of  two 
numbers  added  to  the  less  is  twelve ;  but  the  sum  of 
one  Tourth  of  both  is  only  four.  What  are  the  num- 
bers ? 

16.  If  from  twice  the  equation  2  a; -(-?/=  17,  the 
equation  3  a; -f-^  2/ =  27  be  subtracted,  what  equation 
will  express  the  difference  ?  What  will  be  the  re- 
spective values  of  x  and  y  1 


^25.1  INTELLECTUAL     ALGEBRA.  127 

17.  If  twice  the  equation   —4--^=::  11    be  taken 

from  x-(-y  =  30,  what  will  be  the  difference?  and 
what  the  values  of  x  and  y  respectively  ? 

18.  If  the  equation   — -|- -^  =:  0  be  multiplied  by 

b  3 

3,  and  the  product  subtracted  from  the  equation  x-\- 
y=24,  what  equation  will  represent  the  remainder? 
and  what  will  be  the  respective  values  of  x  and  y  ? 

19.  If  the  ecjuation   -- -|- -  -  =  3   be  multiplied  by 

4,  and  the  product  be  taken  from  the  equation \- 

y 

—  zr:  16,  what  will  represent  the  remainder?     What 

2 

will  be  the  value  of  x,  and  what  of  ?/  ? 

20.  Multiply  the  equation  — -[- —  rz:  5  by  4,   from 

the  product  subtract  the  equation 1-?/=14,  and 

what  will  express  the  remainder?  What  will  be  the 
respective  values  of  x  and  y1 

2  .-r        2  y 

21.  If  three  times  the  equation j =  8    be 

9  .T        2  V 

taken  from  the  equation  -— -| =  33,  what  equa- 
tion will  result  ?  What  will  be  the  value  of  x,  and 
what  of  y  ? 

22.  If  the  equation  -^ — [ =  4    be  multiplied  by 

4,  and  the  product  be  subtracted  from  —^-\-2y:r.i  18, 

vvhat  equation  will  result  ?  and  what  will  be  the 
respective  values  of  x  and  y  ? 


*28  INTELLECTUAL     ALGEBRA. 


SECTION   XXVI. 


[§25. 


1.  Thomas  bought  three  apples  and  one  ^^cacli  for 
five  cents.  Again,  at  the  same  rate,  he  bought  six 
ap])les  and  tliree  peaches  for  twelve  cents.  Wha* 
was  the  cost  of  one  of  each  ? 

Let  X  =^  the  cost  of  an  apple, 

and  7/  r=  the  cost  of  a  peach. 

(1.)   By  one  condition  of  the  question,  6x-\-S  i/  =z  12. 

)  By  another  condition, 3x-\-  i/z=.5 

)  Dividing  1st  by  3, 2x-{-i/  =  4 


)   Subtracting  3d  from  2d, x=l 

)  Takinor  3  x  from  each  mem-  )  -       o 

ber  of  2d, i  ^ 

(G.)   Substituting   1,  the  value  of)  -       .^        r» 

^     '  [  .?/  =  o  — 3,or2. 

X,  tor  X,  m  5th,      J 

An  apple  cost  1  cent,  and  a  peach  2  cents. 

2.  There  are  two  numbers,  sucii  that,  if  si.K  times 
he  less  be  added  to  twice  the  greater,  the  sum  will 

oe  thirty-eight,  and  if  twice  the  less  be  added  to  the 
greater,  the  sum  will  be  fifteen.  What  are  the  num- 
oers  1 

3.  If  the  expression   Gx-{-3y   be   divided    by   3, 
what  expression  will  represent  the  quotient  ? 

4.  What  is  one  third  of  the  expression  6  a:  -|-  3  y  ? 

5.  If  the  expression   Qx-\-Gi/   be   divided   by   3, 
what  will   be  the  quotient  ? 

6.  What  is  one  third  of  9x-\-G)/  1 

7.  What  will  be  the  quotient  of  8  i -]- 4  ^   divided 
i»y  4  ? 

8.  What  is  one  fourth  of  8  z  -f  4  _?/  ? 


§26. 


INTELLECTUAL     ALGEBKA.  129 


9.  If  the  expression  4x-[-2_i7,  divided  by  2,  be 
subtracted  from  3a-  -[-y,  what  will  be  the  remainder? 

10.  If  the  expression  5x-\-5i/,  divided  by  5,  be 
taken  from  3x-(-y,  what  will  be  the  remainder? 

11.  If  the  equation  6x-j-2_y  =  28  be 'divided  by 
2,  and  the  quotient  be  subtracted  from  the  equation 
5  a- -j- _y  =  22,.  what  equation  will  represent  the  re- 
mainder ?  What  will  be  the  respective  values  of  -x 
and  7/  ? 

12.  If  one  third  of  the  equation  6x-\-6i/=:^4S 
be  subtracted  from  5  a;  -f-  2  y  irr:  34,  what  equation  will 
represent  the  remainder  ?  and  what  are  the  respective 
values  of  x  and  i/  ? 

13.  If  you  divide  the  equation  4  x -|- 8  y  :=:  20  by 
4,  and  subtract  the  quotient  from  2x-|-2y  rr:  8,  wh;it 
equation  will  express  the  result  ?  What  will  be  the 
value  of  each  of  the  quantities  x  and  y  ?  '■ 

14.  Divide  Sx-\-3  7/z=2i  by  3,  and  subtract  the 
quotient  from  4  a-  -|-  y  =  23.  What  will  be  the  re- 
mainder ?  What  will  be  the  value  of  x,  and  what 
of  y  ? 

15.  A  man  sold  five  bushels  of  Vvheat  and  five  ol 
rye  for  fifteen  dollars;  and  again,  at  the  same  rate, 
two  of  wheat  and  one  of  rye  for  five  dollars.  What 
was  the  price  of  a  bushel  of  each  ? 

16.  Divide  some  unknown  number  into  two  such 
parts,  that,  if  three  tiines  the  greater  be  added  to  the 
less,  the  sum  will  be  seventeen,  and  the  sum  of  four 
times  the  greater  added  to  twice  the  less,  will  be 
twenty-four.  What  are  the  parts  ?  and  what  is  the 
number  ? 

i7.    A  man  bought  a  saddle  and  bridle,  and,  being 
0 


130  INTELLECTUAL     ALGEBRA. 


[§  26, 


asked  what  he  gave  for  each,  replied,  "  If  four  times 
the  price  of  the  saddle  be  added  to  twice  the  price  of 
the  bridle,  the  sum  will  be  forty-eight  dollars ;  and  if 
three  times  the  price  of  the  saddle  be  added  to  the 
price  of  the  bridle,  the  sum  will  be  ihirty-four '' 
What  did  he  pay  for  each  ? 

IS.  A  farmer,  being  asked  how  many  cows  and 
sheep  he  had,  replied,  "  Two  fifths  of  my  cows  and 
two  thirds  of  my  sheep  would  be  ten  ;  but  one  third 
of  ray  sheep  and  the  whole  of  my  cows  would  be 
thirteen."     How  nxiny  had  he  of  each? 

19.  There  are  two  numbers,  such  that,  if  three 
^ourths  of  the  greater  be  added  to  six  fifths  of  the 
smaller,  the  sum  will  be  twenty-four ;  but  if  two  fifths 
of  the  smaller  be  added  to  one  half  of  the  greater,  the 
sum  will  be  only  one  half  as  much.  What  are  the 
numbers  ? 

20.  If  six  fiftlis  of  Daniel's  age  be  added  to  three 
halves  of  Levi's,  the  sum  will  be  twenty-seven  years ; 
and  if  the  whole  of  Levi's  age  be  added  to  three 
fifths  of  Daniel's,  the  sum  will  be  fifteen.  What  is 
the- age  of  each? 

21.  Divide  twenty-three  into  two  such  parts,  that, 
if  three  sevenths  of  the  greater  be  added  to  two 
thirds  of  the  smaller,  the  sum  will  be  twelve.  What 
are  the  parts? 

22.  A  man,  being  asked  what  he  gave  for  his  horse 
and  cow,  answered,  "  Four  sevenths  of  the  cost  of  the 
horse,  added  to  eight  ninths  of  the  cost  of  the  cow, 
will  be  fifty-two  dollars ;  and  two  ninths  of  the  cost 
of  the  cow,  added  to  two  sevenths  of  the  cost  of  the 
horse,  will  be  twenty-two  dollars."  What  was  the 
cost  of  each  ? 


^26.] 


INTELLECTUAL     ALGEBRA.  131 


23.  If  you  divide  tlie  expression  4  x  -|-  —  by  2,  and 

subtract  the  quotient  from  3  x-|- —,  what  will  repre- 
sent the  remainder? 

24.  What  will  remain  after  subtracting  one  half  of 

4x_L.il  from  5x4-^? 
'3  3 

6  x        3  w 

25.  If  you  divide  —  +  7^  by  3,  and  subtract  the 

quotient  from  —  -(-  —,  what  will  express  the  remain- 
der? 

G  .T        3  v  2  a; 

26.  If  you  take  one  third  of  -f-  -(-  -^^  from  ^  -f  y, 

what  will  express  the  remainder  ? 

27.  If  you  divide  the  equation 
what  equati(*n  will  express  the  quotient  ? 

28. 
=  24? 

29.  If  you  divide  the  equation  —  -f-  -7-  =  20  by 

4,  and  then  multiply  the  quotient  by  3,  what  equation 
will  represent  the  result  ? 

8  X        4  V 

30.  What  is  three  fourths  of  the  equation 1 

5  5 

==32? 

31.  If  you  divide  the  equation 1 r:- 10  by 

2,  and  then  subtract  the  quotient  from \-  y  =z6, 

what  equation  will  represent  the  remainder?     What 
will  be  the  respective  values  of  x  and  y  ? 

32     If  one  third  of  the  equation  3  x  -1-  ^  —  27  bt 


27.    If  you  divide  the  equation  4  x  +  ^  =r  20  by  2, 


28.    What  is  one  fourth  of  the  equation  ~-\-  — 


132  INTKIvLECTUAL     ALGEBRA.  [§  27 

subti  acted     from    7 -\ ^  r:  12,    what     equation    will 

express  the  remahidcr  ?  and  what  will  be  the  re^- 
spective  values  of  x  and  1/  ? 

33.    If  from    22--|-^  — 24    three   fourths  of  the 

equation  x-\-i/=:l7  be  taken,  what  equation  will 
represent  the  remainder?  What  will  be  the  value 
of  X?    and  what  of  y? 


SECTION   XXVII. 

1.  Five  lemons  and  two  pears  were  bought  for 
seventeen  cents.  At  the  same  rate,  four  pears  would 
cost  fourteen  cents  less  than  six  lemons.  What  did 
one  of  each  cost? 

Let  x=z  the  price  of  a  lemon, 

and  1/  =z  the  price  of  a  pear. 

(1.)  By  one  condition  of  the  question,  62:  —  4?/:=  14 

(2.)  By  another  condition, 5  x-\-2i/  =il7. 

(3.)  Dividing  1st  by  2, 'Sx  —  2ij  =  7. 

(4.)  Adding  3d  to  the  2d, 8z  =  24. 

(5.)  Dividing  4th  by  8.^ x  =  2. 

(G.)  Taking  5  a;  from  2d, 2^  =  17  — 5  r. 

(7.)  Substituting  3,  the  value  j.  o  „  _  17  _  15  or  o 
of  X,  in  the  Cth,  ....  J  *       ' 

(8.)  Dividing  7th  by  2,  .  .  .'. y  =  1- 

A  lemon  cost  three  cents,  and  a  pear  one  cent. 

2.  There  are  two  numbers,  such  that,  if  three 
times  the    jxreater  be    added  to  nine  times  the  less, 


§27.] 


INTELLECTUAL     ALGEBRA.  133 


(he  sum  will  be  thirty-six ;  and  if  three  times  the  less 
be  taken  from  four  times  the  greater,  the  remainder 
will  be  eighteen.     What,  are  the  numbers?-^ 

3.  Four  times  John's  money  is-,  six  dollars,  more 
thiMi  twice— Mary's,  .and  the  two  together  have  nine 
dt^/flars.     How  many  dollars  has  each  ? 

4.  If  the  expression  4  *  -[-  4  y  be  divided  by  4, 
and  the  quotient  added  to  2x  —  3',  what  will  express 
the  sum  ? 

5.  If  the  expression  8z-|-2y  be  divided  by  2,  and 
the  quotient  be  added  to  the  quotient  of  6x  —  3y 
divided  by  3,  what  will  represent  the  result  ? 

6.  If  the  equation  6  x -|- 3  ?/ rr:  24  be  divided  by 
8,  and  the  quotient  be  added  to  ^x  —  yz^l,  what 
equation  will  express  the  result  ?  and  what  will  be  the 
respective  values  of  z  and  y  1 

7.  If  the  equation  G  x  —  4y=lG  be  divided  by 
2,  and  the  quotient  be  added  to  3x-(-2?/=  16,  what 
equation  will  result  1  and  what  will  be  the  values  of 
X  and  y,  respectively  ? 

8.  The  difference  between  six  times  Sarah's  age 
and  three  times  Eliza's  is  eighteen  years,  and  the 
sum  of  their  ages  is  one  half  of  the  above  difference. 
What  is  the  age  of  each  ? 

9.  Find  two  such  numbers,  that  if  four  times  the 
less  be  taken  from  eight  times  the  greater,  the  re- 
mainder will  be  twelve,  and  the  sum  of  three  times 
the  greater  added  to  the  less  will  be  seven.  What 
are  the  numbers  ? 

10.  A  man  sold  nine  sheep  and  six  calves  for  forty- 
eight  dollars,   and  he  received  four  dollars   less   for 


IS4  INTELLECTUAL  ALGEBRA.        [§  27. 

three  sheep  than  for  two  calves.     What  did  he  obtain 
for  one  of  each  ? 

11.  There  are  two  numbers,  such  that  the  sum  of 
five  times  the  greater  added  to  ten  thirds  of  the  less 
IS  fifty-five,  and  if  two  thirds  of  the  less  be  subtracted 
from  the  whole  of  fhe  greater,  the  remainder  will  be 
three.     What  are  the  numbers  1 

12.  A  man  bought  a  saddle  and  bridle,  and  said 
that  three  times  the  cost  of  the  saddle,  added  to  three 
fourths  of  the  cost  of  the  bridle,  would  be  fifty-seven 
dollars,  and  that,  if  one  fourth  of  the  cost  of  the  bridle 
be  taken  from  twice  the  cost  of  the  saddle,  the  dif- 
ference would  be  thirty-two  dollars.  What  did  he 
give  for  each  ? 

13.  Find  two  such  numbers,  that  v,hen  four  times 
the  less  is  added  to  four  fifths  of  the  greater,  the 
sum  will  be  twenty,  and  when  one  fifth  of  the  greater 
is  taken  from  the  less,  the  remainder  shall  be  one. 
What  are  the  numbers  l 

14.  A  man  said,  that  the  sum  of  four  fifths  of  the 
value  of  his  horse,  added  to  two  thirds  of  the  value  of 
his  cart,  was  forty-two  dollars,  and  that  the  difference 
between  one  third  of  the  value  of  his  cart  and  three 
fifths  of  the  value  of  his  horse,  was  nineteen  dollars. 
What  was  the  value  of  each  1 

15.  Divide  some  number  into  two  such  parts,  that, 
if  six  sevenths  of  the  greater  be  added  to  three  filths 
of  the  less,  the  sum  shall  be  eighteen,  and  if  one  fifth 
of  the  less  be  taken  from  five  sevenths  of  the  greater, 
the  remainder  shall  be  eight.  What  is  the  number  "^ 
and  what  are  the  parts  ? 


^1  27.1  INTELLECTUAL     ALGEBRA.  135 

16.  A  chaise  and  harness  were  sold  at  such  prices, 
that  if  from  three  fourths  of  the  price  of  the  chaise 
three  fifths  of  the  price  of  the  harness  be  taken,  the 
difference  will  be  sixty  dollars,  and  one  fifth  of  the 
price  of  the  harness,  added  to  one  half  of  the  price 
of  the  chaise,  will  be  fifty-five  dollars.  For  how 
much  was  each  sold  ? 

17.  If  the  expression  9x-\-6  7/  be  divided  by  3, 
and  the  quotient  added  to  x  —  2i/,  what  will  be  the 
sum  ? 

18.  What  is  one  third  of  the  expression  9  x-\-6  i/l 

19.  What  is  two  thirds  of  the  same  expression  ? 

20.  What  is  one  tenth  of  the  same  expression  ? 

21 .  What  is  three  tenths  of  the  same  expression  ? 

22.  If  the  equation   -—  -f-  -^  =  42  be  divided  by 

7,  what  equation  will  represent  the  result  ? 

8  X        12  m 

23.  What  is  one  fourth  of  the  equation 1 Z 

=  40? 

24.  What  will  represent  tliree  fourths  of  the  same 
equation  ? 

25.  If  the  equation ^^=zl8  be  divided  by 

4  ^       y 
2,  and  the  quotient  added  to  the  equation  —  -| = 

27,  what  will  represent  the  sum?    and  .what  will  be 
the  respective  values  of  x  and  y  ? 

20.    If  three  fourths  of  the  equation  x  —  y  ::=:  4  be 

added  to 1 =^r=:21,  what  equation  will  represent 

the  sum  ?     What  will    be  the  value  of  x,   and   whal 
of  y  ?         - 


1 36  INTELLECTUAI,      ALGEBRA. 


SECTION   XXVIII. 


t§2.sj 


1.  A  MAN  weighed  two  kinds  of  cannon  balls  of 
different  weights.  When  he  put  into  the  scale  one  of 
the  heavier  and  two  of  the  lighter,  it  took  seven  one- 
pound  weights  to  balance  them  ;  but  two  of  the 
heavier  and  one  of  the  lighter  weighed  eight  pounds 
What  was  the  weight  of  one  of  each  kind  ? 

Let  z  =  the  heavier  ball, 

and  y  =  the  lighter  ball. 

(1.)   By  one  condition  of  the  question,  .  a--}-2y=n7 

(2.)   By  another  condition, 2x-{-y=r8 

(3.)  Takincr  2  y  from  each  member  )  ^      r> 

^     '  a     J  )  .   .  x=i  t  —  2?/ 

of  1st, )  -^ 

(4.)  Taking  y  from  each  member  of  2d,  2  a;  =  8  —  y 

y 
(5.)  Dividing  4"th  by  2, i  :=  4  —  — 

y 
Since  each    of  the   expressions   7  —  2  3/   and  4 

is  equal  to  x,  they  are  equal  to  each  other,  and 
they  form  a  new  equation,  in  which  y  is  the  only 
unknown  quantity. 

(G.)  Putting  4  —  —,  the  value 

of  7,  for  J,  in  Hd,  .   . 
(7.)  Multiplying  each  mem-)         §_„  _i4_4,, 

ber  of  Gth  by  2,    .   .     )     '  ^  ^' 

(8.)  Adding  4y  — 8  to  each  i  8  — y4-4y  — 8  =  14 

member  of  7th,  .  ..     )      — \y-\-Ay  —  8. 
(9.)  Reducing  8th  by  uniting  terms,  .  ...     3^  =  0. 
(10.)  Dividing  each  member  of  9th  by  3,  .   .    7/ -^  2. 


y  _- 


4-^  =  7-2y. 


§  2S.1  INTELLECTUAL     ALGEBRA.  137 

(11.)  Putting  2,  the  value  of  (  -y       «         o 

1/,  for  1/,  in  the  3d,  .    ) 

Therefore,  one  ball  weighed  3  lbs. ; 
.    the  other  ball  weighed  2  lbs. 

2.  There  are  two  numbers,  such  that  the  sum  of 
twice  the  greater  added  to  three  times  the  less  is 
eleven,  and  three  times  the  greater  is  ten  more  than 
twice  the  less.     What  are  the  numbers  ? 

Let  X  =  the  greater  number, 

and  y  =  the  less  number. 

(1.)  By  one  condition  of  the  question,  2x-\-3i/  =  11. 

(2.)   By  another  condition, 3  a;  m  2  ?/ -j- 10 

(3.)   Subtracting  3y  from  each  )  ^    ,,       ^ 

member  of  1st,  .   .   .  .     ) 
(4.)   Dividing  each  member  of  )  U—3y 

3d  by  2, i    '  '  '  '    ^~      2 

(5.)  Dividing  each  member  of  {  2j/+_l0 

2d  by  3,  ; i    '  '  '  '    '^~      3 

(G.)  Putting  value  of  x  in  5th  )         Sy+jO  _  11  — 3j, 

equaltovalueof  xin4th,  )  3  2 

(7.)   Multiplying  each  member  )  ^^_^ 20  —  33 _9y. 

of  6th  by  6,  and  reducing,  ) 
(8.)   Adding  9?/,  and  subtracting  20  in  7th,  13?/=:  13. 

(9  )  Dividing  by  13  in  8th, y  =  1- 

(10.)  Putting  1,  the  value  of  y,  )  ^_H— 3   ^^  ^ 

for  y,  in  4th, S    *  '  ^  ~     2     '  "'' 

Therefore,  the  greater  number  is  4, 
and  the  less  number  is  1. 

3.  Jf  Sy  be  subtracted  from  each  side  or  member 
of  the  equation  "^  %  ■\- ^  y  r:^  V^ ,  what  equation  will 
lepresent  the  result?  ^^         y«6  •»  <• 


138  INTELLECTUAL      ALGEBRA. 


[§2S. 


4.  If  the  equation  '>ix=  19  —  5?/  be  divided  by  3, 
what  equation  will  express  the  quotient  ? 

5.  What  must  be  done  to  the  equation  3x-\-Iji/  =z 
19,  that  the  value  of  x  may  be  found  in  terms  of  the 
equation  ?  and  what  expression  will  be  equivalent  to 
X,  or  represent  the  value  of  x  ? 

29 5  w 

6.  If   a;  = ',   what   will    represent    twice    xl 

What  six  times  x  ? 

7.  In  the  equation  3  x  -j-  5y  =  19,  what  expression 
will  represent  the  value  of  y  ?     What  of  3?/? 

8.  In  the  equation  ox  —  y:=  13,  if  y  be  added  to 
each  member,  what  equation  will  express  the  result  ? 

9.  If  the  equation  ox=^13-\-i/  be  divided  by  5, 
what  equation  will  represent  the  quotient  ? 

10.  What  must  be  done  to  the  equation  5x  —  y  :^- 
13,  that  the  value  of  x  may  be  found  in  terms  of  the 
equation  ?     What  expression  will  be  equivalent  to  x  ? 

11.  In  the  equation  4x-|-3y  =  22,  what  expres- 
sion will  represent  the  value  of  x  ?  what  the  value 
of  2x? 

12.  In  the  same  equation,  what  will  express  the 
vilue  of  y  ?   what  the  value  of  6y  ? 

13.  Find  expressions  for  the  value  of  x  m  the 
equation  3x-|-2yr=16,  and  in  the  equation  2x-|- 
5  y  =  18.     What  are  the  expressions  ? 

14.  In  the  new  equation  formed  by  these  equivalent 
expressions,  what  will  be  the  value  of  y  ? 

15.  If  you  substitute,  in  the  equation  2  x -\- ii  i/ =: 
18,  the  value  of  y,  thus  foiiiul,  in  the  place  of  y,  what 
will  be  the  value  of  j  ? 


^  2S.  I  INTELLECTUAL     ALGEBRA.  139 

16.  From  the  equations  4  a;  -|-  3  ?/  r=  26,  and  3  z  — 
y  =■  13,  make  an  equation  which  shall  not  contain  the 
quantity  x.     What  will  represent  that  equation  ? 

17.  In  the  new  equation  thus  formed,  what  will  be 
the  value  of  ?/  ? 

18.  If  the  numerical  value  of  i/,  thus  found,  be 
substituted  for  y  in  the  equation  Ax-\-3y  =^ 26,  what 
will  be  the  value  of  x,  after  the  equation  is  reduced  1 

19.  If  twice  John's  age  be  added  to  three  times 
Peter's  age,  the  sum  will  be  thirty-one  years,  and  if 
twice  Peter's  be  added  to  three  times  John's,  the  sum 
will  be  thirty-four.     What  is  the  age  of  each  ? 

20.  There  are  two  numbers,  such  that  three  times 
the  greater  is  one  more  than  five  times  the  less ;  and 
if  twice  the  greater  be  added  to  three  times  the  less, 
the  sum  will  be  twenty-six.     What  are  the  numbers? 

21.  A  boy  bought  four  oranges  and  three  pears 
for  fifteen  cents  :  and  again,  at  the  same  rate,  he 
bought  two  oranges  and  five  pears  for  eleven  cents. 
What  was  the  price  of  one  of  each  ? 

22.  The  sum  of  two  numbers  is  thirteen ;  and 
their  difference  is  seven.     What    are    the  numbers  ? 

23.  If  a  slate  and  two  writing-books  cost  twenty- 
five  cents,  at  the  same  rate  two  slates  would  cost 
fifteeif  cents  more  than  three  writing-books.  What 
was  the  cost  of  one  of  each? 

24.  There  are  two  numbers,  such  that,  if  the 
greater  be  added  to  twice  the  less,  the  sum  will  be 
eleven  ;  and  if  one  be  added. to  three  times  the  less, 
ihe  sum  will  be  twice  the  greater.  What  are  the 
numbers? 

25.  A  man  sold  a  cow  and  a  pig,  at  such  prices 


140  INTELLECTUAL  ALGEBRA.       [§  2S 

that,  if  one  fourth  of  the  price  of  the  cow  be  added 
to  one  half  of  the  price  of  the  pig,  the  sum  will  be 
eight  dollars;  and  if  the  price  of  the  pig  be  taken 
from  five  eighths  of  the  price  of  the  cow,  the  remainder 
will  be  eleven  dollars.  What  was  the  price  of  each? 
Let  %  ^  the  price  of  the  cow, 
and  y  =  the  price  of  the  pig. 
(1.)  By  one  condition  of  the  question,  .  —-[-  —  =  8. 

5  X 
(2.)  By  another  condition, — y=:^\\ 

(3.)  Taking  —  from  each  member  of  1st,  —  =  8  —  - 

3  a; 
(4.)  Adding  y  to  each  member  of  2d,     —zi^\\-\-y 

(5.)  Dividing  each  member  of  4th  by  5,    —  =  — - — 

(C.)  Multiplying   5th    by   2,    and  |  ^        22  +  2 « 

2  X         x  /    •    •  — ^=- 

smce  —  r=  — , I  4.  3 

8  4'  ) 

(7.)  Putting  the  value  of  -  in  3d  |  ^~  +  ~y  _^_  y 

and  6th  equal  to  each  other, -^ 
^8.)  Multiplying  each  member  of  )  44-j-4y=80  — 

7th  by  10,  and  reducing,  .   J      5y. 
(9.)  Adding    5y  —  44    to    each)  „     o^ 

member    of   8th, )     '  '  ' 

(10.)  Dividing  9th  by  9, y  =  4. 

(11.)  Putting  4,  the  value  of  7/,  )  x        „       4         „ 

^  •     <. .  -''S  — =  8 ,orG. 

for  y,  m  3d, )   4  2  ' 

(12.)  Multiplying  11th  by  4, x  =  24 

The  cow  cost  twenty-four  dollars  ; 

the  pig  cost  four  dollars. 

26    There  are  two  numbers,  such  thnt  i^  'mv;  Ljilru^t 


§2S.J 


INTELLECTUAL      ALGEBRA.  141 


29.    If  the  equation  —  =  6 be  multiplied  by 

3  8 


of  the  greater  be  added  to  two  thirds  of  the  less,  the 
sum  will  be  ten  ;  and  if  one  third  of  the  less  be  added 
to  three  fourths  of  the  greater,  the  sum  will  be  ten. 
WJjat  are  the  numbers  ? 

27.  In  the  equation  -—-\ ^  =  12,  if  —  be  sub- 

'  3      '      4  '4, 

iracted  from  each  member,  what  equation  will  repre^ 
sent  the  remainder  ? 

28.  If  the  equation  —  =  12 be  divided  by  2, 

what  equation  will  express  the  quotient  ? 

29.  If  the  equation  — m  6  — 
3,  what  will  express  the  product  ? 

30.  What  must  be  done  to  the  equation  —  ~\ 

rr:  12,  that  the  value  of  x  may  be  found  in  terms  of 
the  equation?  What  expression  will  be  equivalent 
to  xl 

31.  If  x==  18  — ^,  what  will  3r  equal '? 

2.  X        3  V 

32.  In  the  equation [ — -=:  12,  what  expression 

will  represent  the  value  of  y  ? 

33.  What  will  represent   the  value  of  2y,  in  the 

same  equation  ? 

3  X         V 

34.  What  must  be  done  to  the  equation —  = 

7,  that  an  expression  may  be  found  equivalent  to  a;  ? 

35.  What  is  the  expression  for  the  value  of  x,  in 
the  last  equation  ? 

3G.    Find  equivalent  expressions  for  the  value  of  x, 

1  •  ~  ^    ,      V  -.rw  ,    3  X  y  _ 

in    the    equations U  —  =  10,    and —  =  5. 

*  3      '      4  4  2 

What  will  the  expressions  be  ? 


142         INTELLECTUAL  ALGEBRA.       T^  29 

37.  In  the  new  equation  formed  by  the  \<Jiies  of  a; 
V  is  tlie  only  unknown  quantity,  and  what  is  its  value? 

38.  If  you  substitute,  In  the  equation —  =  5, 

the  value  of  y,  as  found  above,  in  the  place  of  y,  wfiat 
will  be  the  value  of  x  ? 

39.  'From  the  equations ^  :=z  6,    and    — + 

*  2  3'  4     ' 

—  z=.l.  make  an  equation  which  shall  not  contain  the 

unknown  quantity  x.     What  will  that  equation  be? 

40.  In  the  new  equation  thus  formed,  what  will  be 
the  value  of  y  ? 

41.  If  the  value  of  y,  thus  found,  be  substituted 

for  y,  in  the  equation —  r=  6,  what  will  be  the 

»  2  3 

value  of  a;  ? 

42.  Two  boys  bought  a  dog  for  six  dollars.  John 
says,  "I  will  give  one  half  of  my  money,  and  you 
can  give  one  third  of  yours,  and  that,  will  just  pay  for 
him,  but  I  shall  own  the  greater  part  of  him."  "  No," 
says  Henry,  "  I  will  give  two  thirds  of  my  money,  and 
you  shall  give  only  one  fourth  of  yours :  that  will  pay 
for  him,  and  I  shall  own  more  than  you."  How  much 
money  had  each  ? 


SECTION   XXIX. 

1.    A  MAN  weigher"  two  kinds  of  cannon  balls  of 

different  weights.     Three  of  the  heavier  and  two  of 

the  lighter  weighed  twenty-one  pounds,  while  one  of 


§29-1  INTtLLECTIlAL     ALGEBRA.  143 

ihe    heavier   and   one    of  the  lighter   weighed   eight 

pounds.     What  was  the  weight  of  one  of  each  kind  ? 

Let  X  :=  the  weight  of  one  of  the  heavier  balls, 

and  1/  =  the  weight  of  one  of  the  lighter  balls. 

(1.)  By  one  condition  of  the  question,  3  2;-{-2y  =  21. 

(2.)  By  another  condition, x-\-  7/  :=8. 

(3.)  Taking  x  from  each  member  of  2d,  .  y  =  8  —  x. 

(4.)  Multiplying  3d  by  2, 2i/  =  16—2x. 

(.5.)  Putting  the  value  of  2 y  for)    3  2;-fl6  — 2x 

2i/  in  1st, i  =21. 

(6.)  Taking  16  from  each  member  ) 

of  5th,  and  uniting  terms,   .  J 
(7.)  Putting  5,  the  value  of  x,  in  3d,  y  =  8  —  5,  or  3. 
One  kind  of  ball  weighed  5  lbs; 
the  other  kind  weighed  3  lbs. 
2.   There  are  two  numbers,  such  that  three  times 
the  greater  and  twice  the  less,  when  added,  are  twenty- 
two,  and  twice  the  greater  is  six  more  than  three  times 
the  less.     What  are  the  numbers  ? 

Let  X  =  the  greater  number, 

and  y  z=  the  less  number. 

(1.)  By  one  condition  of  the  question,  3x-|-2y  r=22. 

(2.)  By  another  condition, 2xr=3y-|-6. 

(3.)  Dividing  2d  by  2,  . xr=^-f-3 

(4.)  Multiplying  3d  by  3, 3x  =  ^4-9. 

(5.)  Putting  this  value  for  Sx\9  y  _,      ^^^^     ^^ 

in  1st, ^2 

(G.)  Mukiplying  5th  by  2,  .  .  .  9y  + 18  +  47/ =  44. 
(7.)  Taking  18  from  each  mem-  \ 

ber  of  6th,  and  uniting  > I3y  =  26. 

terras, ■' 


141  INTELLKCTLAL  ALGEBRA.       f'^  29 

(S.)  Dividing  7th  by  13, y  :rr  2 

(9.)  Putting  2,  the  value  of  ?/,  |  c    ,   „        ^ 

m  the  3d, i  2  ^    ' 

The  greater  number  is  G ; 
the  smaller  is  2. 

3.  If,  from  each  member  of  the  equation  3z-|-2y 
r=21,  2  y  be  taken,  what  equation  will  express  the 
remainder  ? 

4.  If  the  equation  3z=z21 — 2j/  be  divided  by 
3,  what  equation  will  express  the  quotient  ? 

5.  What  must  be  done  to  the  equation  3x-|-2y 
^=21,  to  find  an  expression  equivalent  to  x,  or  that 
will  represent  the  value  of  :r  ? 

G.    If  a:  =  7  —  ^,  what  will  2x  equal? 

7.  In  the  equation  2x-\~2^  =^21,  what  expression 
will  represent  the  value  of  y  ? 

8.  In  the  equation  4 1  -j-  3  ?/  =  22,  what  expression 
will  represent  the  value  of  x  1 

0.  In  the  same  equation,  what  expression  will  rep- 
resent the  value. of  y1 

10.  In  the  equation  2x-\-3 y  =:z  12,  find  an  ex- 
pression to  represent  the  value  of  x.  What  will  the 
expression  be  ? 

11.  Substitute  the  value  of  x,  thus  found,  in  the 
place  of  X,  in  the  equation  3 1 -}- 2  y  :=  13.  What 
iftill  the  equation  then  be? 

12.  If  tho  equation  18  — ^-|-2y  =  13  be  mul- 
tiplied by  2,  what  equation  will  express  the  product  7 

13.  In  the  equation  36  —  5y=  26,  if  5  y  be  added 
to  ET^h  member,  what  will  express  the  result  ? 


§  29.]  INTELLl^CTUAL      ALGEBRA,  145 

14.  If  26  be  taken  from  each  member  of  the  equa- 
tion 36  =1  26 -\- 5  1/ ,  what  will  express  the  result?  and 
what  is  the  vakie  of  y  ? 

15.  If,  in  the  equation  2x-^'3  1/  =z  12,  2  be  sub- 
stituted in  the  place  of  y,  what  will  the  value  of  x  be  ? 

If).  A  man  bought  two  barrels  of  cider  and  one  o'" 
apples  for  eight  dollars,  and  found  that,  at  the  same 
rate,  three  barrels  of  cider  would  cost  five  dollars 
more  than  two  barrels  of  apples.  What  did  a  barrel 
of  each  cost  ? 

17.  There  are  two  numbers,  such  that  three  times 
the  greater  is  two  more  than  four  times  the  smaller, 
and  five  times  the  smaller  is  eight  more  than  twice 
the  greater.     What  are  the  numbers  ? 

18.  Twice  John's  money  is  twelve  dollars  more 
than  Henry's;  and  twice  Henry's  is  six  dollars  more 
than  John's.     How  many  dollars  has  each  1 

19.  If  you  add  four  to  the  numerator  of  some  frac- 
tion, the  fraction  will  be  one  half;  but  if  you  add 
four  to  the  denominator,  the  fraction  will  be  only  one 
fourth.     What  is  the  fraction  ? 

Let  X  z=  the  numerator, 
and  y  =:  the  denominator  ; 

then  —  =.the  fraction. 
y 

(1.)  By  one  condition  of  the  question,  .  .   — !— z= — 

y         2 

(2.)  By  another  condition,     -zr-  —. 

1/-H4  4 

(3.)  Multiplying  1st  by  y, x-f_4r=|-. 

(4.)  Multiplying  3d  by  2,      2^4- 8  =  3^^. 

10 


146  INTELLECTUAL     AJ.GEBRA.  [§  29 

4  X 

(5.)  Multiplying  2d  by  4, — —  = 

(6.)  Multiplying  5th  by  J/ -}~ 4,    .  .      .    Ax  =  y  -\-4 
(7.)  Dividing  6th  by  2, T.  .    2x  =  -^4-2. 


(8.)  Putting  this   value  of  2  x   in  )  J/^   i   o   i   e 

the  4th ^s""""^     ~~^ 

lOrrry  — 


the  4th, 
(9.)   Subtracting    -^,  and    uniting  I  y 


9 

terms    in   8th, 

(10.)  Multiplying  by  2, 20  =  2y  —  y,oTy. 

(11.)  Subtracting     4    from    each  |  y_ . 

member    of    3d, )  2 

(12.)  Putting  20,  the  value  of  y,  V     _20 ,         ^ 

in  11th, K"  2  '  '^'"    ^ 

6=  numerator, 
20  :=:  denominator ; 

the  fraction  is  — . 

20 

20.  Mary  says  one  third  of  her  age  and  four  years 
more  are  equal  to  one  half  of  Susan's  age ;  and  Susan 
says  one  fourth  of  her  age  and  six  years  more  are 
equal  to  five  sixths  of  Mary's.     How  old  is  each? 

21.  If  you  subtract  three  from  the  numerator  of 
some  fraction,  the  fraction  will  be  equal  to  one  sev- 
enth ;  but  if  you  subtract  three  from  the  denominator, 
the  fraction  will  be  equal  to  one.  What  is  the  frac- 
tion? 

22.  If  the  equation  —  =  — -f-4   be  multiplied  by 

3  4 

3    and    divided    by  2,  what   equation  will    represent 
the   result  ? 

23.  If    4    be    taken    from    eacli    member    of    tl:e 


§  30.]  INTELI.KCTUAL     ALGEBRA.  147 

equation —^  r= 1-4,  and   the  remainder  multiplied 

by  4,  what  expression  will  represent  the  value  of  y  ? 

X         2  ?/ 

24.  In  the  equation  —  rr:  ^  —  2  find  the  value  of 

X,  and  substitute  it  for  x,  in  the  equation  2a;  =  — -|- 

6;   and  then  find  the  respective  values  of  x  and  1/  in 
numbers.     What  are  they  ? 

25.  Samuel  and  Nathan  have  seven  dollars.  Half 
of  Samuel's  money  is  one  dollar  more  than  one  third 
of  Nathan's.     How  many  dollars  has  each? 


SECTION   XXX. 

1.  When  a  number  is  multip.'ied  by  itself,  the 
product  is  called  the  second  poiver  or  square  of  tliat 
number,  and  the  number  itself  is  called  the  second 
root  or  square  root  of  that  square  or  product.  Thus, 
2  X  2  =  4 ;  and  4  is  the  square  or  second  power  of 
2,  and  2  is  the  square  root  or  second  root  of  4. 

2.  What  is  the  square  or  second  power  of  three  ? 

3.  What  is  the  square  root  or  second  root  of  nine  ? 

4.  What  is  the  second  power  or  square  of  four  1 

5.  What  is  the  second  root  or  square  root  of  six- 
teen ? 

G.    What  is  the  second  power  or  square  of  six? 

7.  What  is  the  second  or  square  root  of  forty-nine  1 

8.  The  square  or  second  power  of  x  is  x  times  i, 
which  maybe   expressed   thus;  r  3-,  or  r^;    and  th 


148         INTELLECTUAL  ALGEBRA.       [§  30. 

square  root  or  second  root  of  x^  is  x  ;  or,  using  the 
radical  sign,  is/x~z=.%.     The  square   of  3  a;  is  9  x^, 

and  the  square  root  of  9  x-  is  3  x,  or  \/  ^  ^^  =  3  x. 

9.  What  is  tlie  square  or  second  power  of  y  ? 

10.  What  is  the  square  root  or  second  root  of  y-? 

11.  What  is  the  square  or  second  power  of  2?/? 

12.  What  is  the  second  or  square  root  of  4  y^  ? 

13.  What  is  the  square  or  second  power  of  5x  ? 

14.  What  is  the  second  or  square  root  of  3Gx2? 

15.  How  many  times  is  x  contained  in  x  ? 

16.  If  X  be  divided  by  x,  what  will  the  quotient  be  ? 

17.  If  1  be  multiplied  by  x,  what  will  the  product 
be? 

18.  If  2  be  multiplied  by  x,  what  will  the  product 
be? 

19.  If  2  be  multiplied  by  x^,  what  will  the  product 
be? 

20.  If  2  X  be  divided  by  x,  what  will  be  the  quo- 
tient ? 

21.  If  2x  be  divided  by  2,  what  will  the  quotient 
be? 

22.  If  7  X  be  divided  by  x,  what  will  the  quotient 
be? 

23.  If  4  X  be  multiplied  by  2,  what  will  the  product 
be? 

24.  If  8  X  be  divided  by  4  x,  what  will  be  the  quo- 
tient ? 

25.  If  2  X  be   multiplied   by  x,  what  will   be  the 
product  ? 

26.  If  2  X    be    multiplied    by  2  x,   what    will    the 
product  be  ? 


^  30.]  INTKLLECTUAL     ALGEBRA.  149 

27.  If  4  X-  be  dinded  by  2  x,  what  will  the  quotient 
be? 

28.  If  4  x^  be  divided  by  4  x,  what  will  the  quotient 
})e? 

29.  What  is  the  second  power  of  3  a-  ? 

30.  What  is  the  square  root  of  9i-? 

31.  If  X  =  3,  to  what  will  x^  be  equal  ? 

32.  If  a;2  —  9,  to  what  will  x  be  equal  ? 

33.  If  X  =r  4,  to  what  will  z^  be  equal  ? 

34.  If  x2  =  16,  to  what  will  x  be  equal  1 

35.  If  a;  =  5,  to  what  will  x^  be  equal  ? 

36.  If  x~  =  36,  what  will  x  equal  ? 

37.  If  2  z^  =  18,  what  will  x^  equal  ? 

38.  If  the  equation  2  a;^  —  18  be  divided  by  2,  what 
equation  will  represent  the  quotient  ? 

39.  If  the  equation  3  x^  =  12  be  divided  by  3,  what 
equation  will  express  the  quotient?  What  will  be  the 
value  of  X  ? 

40.  Find  the  square  root  of  each  member  in  the 
equation  x^  =z  16.     What  is  the  value  of  x  ? 

41.  Find  the  square  root  of  each  member  of  the 
equation  4  x^  =  36.     What  will  be  the  value  of  x  ? 

'2,  X  .     2.  X  .     .  2  :b 

42.  The  square  of  ^^  is  —  multiplied  by  — ,  which 

4  x'*  4  x^  i 

IS  —  ;  and  the  square  root  of  —  must  therefore  be 

9  ^  9 

^^.     What  is  the  square  of  —  ? 

x^ 

43.  What  is  the  square  root  of  — ? 

4 

44.  What  is  the  square  of  — ? 

^  3 


150  INTELLECTUAL  ALGEBRA.        [§  31. 


45.  If  —    be  multiplied  by  — ,  what  will  be  the 

o  3 

product  ? 

4ar2 

46.  What  is  the  square  root  of  — ? 

*  25 

47.  What  is  the  square  of  —  ? 

2  X  -  2  X 

48.  If  —  be  multiplied  by  ^  what  will  be  the 


product  ? 
49.    W 

.50.    To  what  is  the  expression  \/    — ^  equal  ? 


49.    What  is  the  second  root  of ? 

6-1 


SECTION  XXXI. 

1.  John  says,  if  his  age  were  multiplied  by  his 
age,  the  product  would  be  five  times  his  age.  How 
old  is  he  ? 

Let  X  represent  John's  age ; 
tlien  z  X  3^  =  2;-,  the  product  of  his  age  multiplied  by 

his  age. 

But  this  product  must  be  equal  to  5  times  his  age 

or  5  z  ; 

therefore,  by  a  condition  of  the  question, 

z'2  =  5x. 

Dividing  this  equation  by  x,  gives 

z  =  5; 


§31.]  INTELLECTUAL     ALGEBRA.  151 

for,  if  z  X  2;  is  equal  to  5  times  x,  or  x  times  5, 

then,  z  not  multiplied  by  x,  must  be  equal  to  5  not 

multiplied  by  x ; 

and  the  age  of  John  is  5  years. 

2.  What  number  multiplied  by  itself  will  be  six 
times  the  number  1 

3.  A  father  is  six  times  as  old  as  his  son,  and  his 
age  is  the  square  of  his  son's  age.  What  is  his  son's 
age? 

4.  The  square  or  second  power  of  a  number  is 
seven  times  the  number.     What  is  the  number  ? 

5.'   George  has  twice  as  many  cents  as  Robert,  and 
if  Robert's  money  be  multiplied  by  the  number  of 
cents  that  George  has,  the  product  will  be  double  the 
money  that  both  have.     How  many  cents  has  each  ? 
Let  X  z=i  Robert's  money  ; 
.  then  2  a;  =  George's  money, 
and  X  -j-  2  a;  =:  3  X  will  be  the  sum  of  money  that  both 

have, 

and  X  X  2  a;  :=  2  z-  will  be  the  product  of  one's  money 

by  the  other's. 

Therefore,  by  the  conditions  of  the  question, 

2  x2  =  6  X. 

Dividing  this  by  2:c,  gives 

xr=3. 

2  X  ==  2  X  3,  or  6  ; 

then  George  has  C  cents,  and  Robert  3  cents. 

6.  There  is  a  field  which  is  as  many  rods  long  as  it 
is  broad ;  and  twice  the  product  of  its  length  by  its 
breadth  is  equal  to  ten  titpes  its  length.  What  is  the 
length  of  one  side  of  The  field  ? 

7.  Four  times  the  product  of  a  number  multiplied 


152         INTELLECTUAL  ALGEBRA.        [§  31. 

by  itseJf  ii  equal  to  sixteen  tiroes  the  number.     What 
ia  the  number  ? 

8.  Sarah  hzs  three  thnes  as  many  books  as  Jane 
If  Sarah's  number  of  books  be  multipiied  by  Jane's 
number,  the  product  will  be  nine  tir/ies  the -difference 
bettveen  Sarah's  and  Jane's.     How  many  has  each  ? 

9.  One  number  is  twice  as  large  as  another  :  and 
the  product  of  the  two  numbers  is  fourteen  times  their 
difference  ?     What  are  the  numbers  ? 

10.  There  in  a  square  field,  and  its  length  multi- 
plied by  its  breadth,  or  the  number  of  srjuare  rrxJs  in 
the  field,  is  equal  to  the  number  of  rods  round  it. 
How  many  rods  long  is  one  side  of  the  field  ?        /> 

11.  One  number  is  three  times  another;  and  the' 
prr>duct  of  the  two  is  e^]ual  to  three  times  their  sum. 
What  are  the  numbers  ? 

12.  John  has  four  times  as  many  apples  as  William  ;\ 
and  if  John's  number  of  apples  be  multiplied  by  twice! 
WiJiiam's  number,  the  product  will  be  eight  times/ 
what  they  both  had.     How  many  had  each  ?  / 

13.  One  number  is  twice  another;  and  if  three 
times  the  smaller  be  multiplied  by  twice  the  larger, 
the  product  will  be  twenty-four  times  their  difference. 
What  are  the  numbers  ? 

14.  If  2  be  multiplied  by  t,  what  will  be  the 
pro<luct  ? 

15.  If  iz  be  divide/]  by  z,  what  will  be  the  quo- 
tient ? 

16.  If  2z  be  divided  by  2,  what  will  be  the  quo- 
tient ? 

17.  If  6z  be  divided  by  -ix,  what  will  be  the 
quotient  ? 


5^31.]  IXTKLLECTITAI-     ALGEBRA.  153 

IS.  If  4  r'*  ho  divided  by  j,  what  will  bo  tho 
quotient  ? 

19.  If  tho  equation  3=  4  bo  multiplied  by  r,  what 
equation  w  ill  express  the  pnxluct  ? 

'20.  If  4  J-  be  divided  by  4  3,  what  will  he  the 
quotient  ? 

'21.    If  r'^  =  4  r,  what  will  3  oqwal  ? 

'2*2.  If  tho  equation  t  =  4  bo  multiplied  by  *2  t, 
what  will  express  the  prixluct  ? 

'2J^.  If  tho  equation  3-^=4  r  bo  divided  by  3,  what 
equatit-tn  will  express  the  quotient  ? 

'24.  If  tho  e4^u»ti<-»n  '2a:'^r=8  3:  be  divided  by  Hx, 
what  equati(.>n  will  express  tlK>  q<ioticnt  ? 

'25.  If  i»  3-  =:  !,■>  9  he  dividiHl  by  5  T,  what  equation 
will  express  tho  quoueni  ? 

'2(>.  In  tho  eciiiiiiuin  4  J- -;  1-r.  what  number  dix">s 
*  represent 

'27.  Ivoduco  tlio  equation  (.1  j'^;^  1>  J,  and  tind  the 
value  of  r. 

'2S.  A  father  is  live  times  as  old  as  his  son,  and  the 
produet  of  their  aijes  is  five  times  the  sum  of  their 
aires.     What  are  their  ages  ? 

'29.  One  number  is  four  times  another,  and  twelve 
tinier  their  sum  is  equal  to  Uuce  times  their  product. 
What  are  the  numbers  ? 

3l).  A  field  is  three  times  as  many  rods  long  as  it 
is  wide,  and  the  product  of  the  length  by  twice  the 
width  is  eighteen  limes  tho  diflorence  l>etween  tho 
lenijth  and  width.     How  many  rods  long  is  each  side  ? 

;11.  One  number  is  five  times  another,  and  their 
product  is  equal  to  ton  times  their  dilTerence.  What 
are  the  numbers  ? 


154  INTELLECTUAL  ALGEBRA.        [§  32 

32.  A  farmer  has  twice  as  many  oxen  as  horses ; 
and  if  the  number  of  oxen  be  multiplied  by  the  num- 
ber of  horses,  the  product  will  be  twice  the  sum  of 
the  oxen  and  horses  together.  How  many  of  each 
has  he  ? 


SECTION   XXXII. 

1.  A  BOY,  being  asked  his  age,  replied,  "  If  my  age 
were  multiplied  by  one  half  of  my  age,  the  product 
would  be  six  times  my  age."     How  old  was  he  ? 

Let  X  =1  his  age  ; 

then  —  =  one  half  of  his  age, 

and  X  X  —  ^=  IT  ^^'^^  ^^  ^^^  ^S^  multiplied  by  half 

of  his  age. 

By  the  condition^  of  the  question, 

—  =  Q  X. 

2 

Multiplying  each  member  by  2, 

i2  —  12  z. 

Dividing  each  member  by  x, 

z=12; 

therefore  the  boy's  age  was  twelve  years. 

2.  What  number,  multiplied  by  one  third  of  itself, 
will  give  a  product  equal  to  twice  the  number  ? 

3.  If   —    be    multiplied  by  2,   what   will  be   the 

product  ? 

4.  If  2^  be  divided  by  2,  what  will  be  the  quotient ! 


§  32.]  INTELLECTUAL      ALGEBRA.  153 

5.  What  expression  will  represent  one  half  of  a^? 

6.  If  z^  be  divided  by  3,  what  will  express  the 
quotient  1 

7.  What  expression  will  represent  one  fourth  of  x^  ? 

8.  What  expression  will  represent  three  fourths 
of  a;2? 

9.  If  2  x~  be  divided  by  3,  what  expression  will 
represent  the  quotient  1 

10.  If  3  x^  be  divided  by  4,  what  will  express  the 
quotient  ? 

11.  If  3x2  bg  divided  by  2,  what  will  express  the 
quotient  ? 

12.  If  the  equation  x^^^A  be  divided  by  2,  what 
equation  will  represent  the  quotient  ? 

13.  If  the  equation  x^z=z6x  be  divided  by  3,  what 
will  represent  the  quotient  ? 

14.  If  the  equation  2  x^  =  12  x  be  divided  by  3, 
what  will  express  the  quotient  ? 

15.  What  is  one  third  of  the  equation  2x^=12x1 

16.  What  is  two  thirds  of  the  equation  x^  =.6x1 

17.  If  the  equation  —  =r=3x  be  multiplied  by  3, 
what  equation  will  represent  the  product? 

18     If  the  equation  —  :=  6  x  be  divided  by  2,  what 

equation  will  express  the  quotient? 
3  x^ 

19.  In  the  equation  —  =3x,  what  is  the  value 

if  X? 

20.  If  one  fourth  of  George's  money  be  multiplied 
by  the  whole  of  his  money,  the  product  will  be  four 
times  his  money.     How  many  dollars  has  he  ? 

21.  One  mimber  is  one  fifth  of  another,  and  their 


156  INTELLECTUAL      ALGEBRA.  [§32. 

product  is  equal  to  twice  the  greater.  What  are  the 
numbers  ? 

22.  A  man,  being  asked  how  many  children  he  had, 
replied,  that  if  the  number  of  his  children  were  mul- 
tiplied by  itself,  three  fifths  of  the  product  would  be 
six  times  the  number  of  his  children.  How  many 
had  he? 

23.  If  the  square  of  a  number  be  added  to  one 
fourth  of  the  square,  the  sum  will  be  ten  times  the 
number.     What  is  the  number  ? 

24.  Caroline  has  one  third  as  -many  books  as 
Eliza,  and  the  difference  between  the  square  of 
Eliza's  number  and  the  square  of  Caroline's  number 
is  twelve  times  the  number  that  Eliza  has  more  than 
Caroline.     How  many  has  each  ? 

25.  One  number  is  one  half  of  another,  and  the 
sum  of  their  squares  is  ten  times  the  sum  of  the  num- 
bers.    AVhat  are  the  numbers  ? 

•  26.  A  farmer  bought  a  cow  for  one  half  of  what 
he  gave  for  a  horse.  If  the  square  of  the  cost  of  the 
cow  be  subtracted  from  the  square  of  the  cost  of  the 
horse,  the  difference  will  be  twenty-five  times  the  cost 
of  both.     What  did  he  pay  for  each  ? 

27.  One  number  is  one  fifth  of  another,  and  the 
square  of  the  smaller  is  equal  to  the  difference 
between  the  two  numbers.  What  are  the  num- 
bers? 

•  28,  A  received  twice  as  many  dollars  as  B.  If 
A's  money  be  multiplied  by  two  thirds  of  B's,  the 
product  will  be  four  times  the  sum  of  what  they  both 
received.     How  much  money  did  each  receive  ? 

20     One   number    is   two    thirds  of   another,   and 


§  32.]  INTELLECTUAL      ALGEBRA.  157 

their  product  is  three  times  the  less.     Wliat  are  the 
numbers  ? 

30.  A  bridle  cost  one  fourth  as  many  doHars  as  a 
saddle ;  and  if  the  price  of  the  bridle  be  multiplied 
by  the  price  of  the  saddle,  the  product  will  be  the 
price  of  four  saddles.     What  was  the  cost  of  each  ? 

31.  What  is  the  square  of  2  a;  ? 

32.  What  is  the  square  of  —  ? 

2  X 

33.  What  is  the  square  of  — ? 

3  X 

34.  What  is  the  square  of  -7-  ? 

5 

35.  If  x~  be  added  to  — ,  what  will  express  the  sum 

in  one  term  ? 

2  x^ 

3G.    Reduce  the  expression  x^  -\ — 7-  to  one  term. 
87.    If    -  be  taken  from  x^,  what  will  express  the 

remainder  ? 

2  x^ 

38.  If  —  be  taken  from  x'^,  what  will  express  the 

remainder  ? 

x^  3  x^ 

39.  If  —  be  taken  from  ,  what  will  represent 

vhe  remainder  ? 

40.  If  ^^-^  be  subtracted  from  x~,  what  will  express 

5 

he  remainder  ? 

41.  If  the  equation  —  =  x  be  multiplied  by  3,  what 


equation  will  represent  the  product? 

miiltinlipri    hv  4. 

03 


3  X- 
42.    If  the  equation   — ^nGx  be  multiplied  by  4^ 


158  INTELLECTUAL  ALGEBRA.       [§  33. 

and  that  product  divided  by  3  x,  what  equation  will 

express  the  result  ? 

2  x2 

43.  What  must  be  done  to  the  equation  —  =■  2  x 

to  find  the  value  of  a;  ?     What  number  does  x  repre- 
sent'' 

2  X- 

44.  Reduce  the  equation  —  =:  4  x.     What  will  be 

the  value  of  x  ? 


SECTION   XXXIII. 

1.  A  BOY,  being  asked  his  age,  said,  if  his  age 
were  multiplied  by  itself,  the  product  would  be  forty- 
nine.     What  was  his  age  ? 

Let  X  rr  the  boy's  age ; 
then  xy<,x=-x^  will  be  his  age  multiplied  by  itself.  J 
But  his  age  multiplied  by  itself  is  49 ;  ^ 

therefore,  by  the  conditions  of  the  question, 
x2  =  49. 
Extracting  the  square  root  of  each  member,  that  is, 
finding  a  quantity  which,  multiplied  by  itself,  will 
produce  each  member,  gives 

X  r=  7,  the  boy's  age. 

It  is  evident  that  x  is  the  second  root  of  x~,  because 

xXx  =  x^; 

and  7  is  the  second  root  of  49,  because 

7  X  7  =  49. 

2.  The  second  power  or  square  of  a  number  is 
nine.     What  is  the  number  ? 


^  33.]  INTELLECTUAL     ALGEBRA.  1  59 

3.  If  x~  be  divided  by  x,  what  will  be  the  quotient? 

4.  If  forty-nine  be  divided  by  seven,  what  will  be 
the  quotient  ? 

5.  If  the  square  root  of  the  equation  a--  =  49  be 
extracted,  what  equation  will  express  the  result  ? 

6.  If  each  member  of  the  equation  z  =  7  be 
multiplied  by  itself,  what  equation  will  express  the 
product  1 

7.  If  the  equation  2  a;^  =  18  be  divided  by  2,  what 
will  express  the  quotient  ?  and  what  will  be  the  value 
of  x\ 

8.  John  has  twice  as  much  money  as  George,  and 
the  product  of  George's  money  multiplied  by  John's 
is  fifty  cents.     How  many  cents  has  each  ? 

9.  Three  times  the  product  of  a  number  by  itself 
is  forty-eight.     What  is  the  number  ? 

10.  A  man  received  as  many  shillings  a  day  as  he 
.worked  days.  How  many  days  did  he  work  to  obtain 
six  dollars  ? 

11.  Four  times  the  square  or  second  power  of  a 
number  is  one  hundred.     What  is  the  number  1 

12.  There  is  a  square  field  containing  one  hundred 
and  twenty-one  square  rods.  How  long  is  one  side 
of  the  field  ? 

13.  One  half  of  the  product  of  a  number  multi- 
plied by  itself  is  eight.     What  is  the  number  1 

14.  If  each  member  of  the   equation  —  =  8  be 

multiplied     by    2,    what     equation    will     express    tlie 
product  1 

15.  If  each  member  of  the  equation   —  =  24   be 

^  3 


160         INTELLECTUAL  ALGEBRA.        [^  SA 

multiplied     by    3,    what   equation    will    express    the 
product  ? 

10.  If  each  member  of  the  equation  2  x-  =^  72  be 
divided  by  2,  what  will  express  tlie  quotient  ? 

17.  In  the  equation  —  r=  24,  what  number  is 
represented  by  x  ? 

18.  In   the   equation   -;-  =  CO,  what   is  the  value 

of  x? 

19.  Sarah's  age  is  one  half  of  Matilda's,  and  the 
product  of  their  ages  is  thirty-two.  What  is  the  age 
of  each  ? 

20.  If  one  half  of  a  number  be  multiplied  by  one 
liird  of  the  same  number,  the  p'-oduct  will  be  twenty- 
four.     What  is  the  number  ? 

21.  A  field  containing  ninety-six  square  rods  is 
two  thirds  as  wide  as  it  is  long.  What  is  its  length, 
and  what  its  width  ? 

22.  If  one  half  of  a  number  be  added  to  itself, 
and  the  sum  multiplied  by  itself,  the  product  will  be 
ninety-six.     What  is  the  number  ? 

23.  If  —  =  40  be  multiplied  by  2,  what  equation 

will  express  the  product  ? 

24.  If  5x2  =  80  be  divided  by  5,  what  will  be  the 
quotient  ? 

3  .T* 

2.5.    In  the  equation    -^  =  40,  what  is  the  value 

of  x? 

20.  A  farmer  said,  if  one  half  of  nis  sheep  were 
multiplied  by  one  fourth  of  them,  the  product  would 
be  fiftv.     How  manv  had  he  ? 


t,  33.  '  INTb:LLl!;c;TUAL     ALGEBKA.  161 

27.  If  six  be  added  to  the  square  of  a  number,  the 
sum  will  be  fifty-five.     What  is  the  number  1 

28.  If  3  be  subtracted  from  each  member  of  the 
equation  x^  -\-  3  =:  19,  what  equation  will  express  the 
remainder  ?  and  what  will  be  the  value  of  r  ? 

29.  If  7  be  added  to  each  member  of  the  equation 
2 -T^  —  7  =  43,  what  will  express  the  sum?  and  what 
will  be  the  value  of  a;  ? 

30.  A  boy  said,  if  two  cents  were  added  to  twice 
the  square  of  his  money,  he  would  have  one  dollar 
How  many  cents  had  he  ? 

31.  The  product  of  two  numbers  is  sixteen,  and 
the  less  is  contained  four  times  in  the  greater.  What 
are  the  numbers  ? 

32.  A  boy  paid  eighty-one  cents  for  melons,  giving 
for  each  melon  a  number  of  cents  equal  to  the  num- 
ber of  melons  that  he  bought.     How  many  did    he 

buy? 

*'  2 

33.  If  the   equation   —  =  —  be  multiplied  by  6. 

6  3 

what  equation  will  represent  the  product  ?  and  what 
will  be  the  value  of  a;  ? 

34.  Since  the  square  root  of  four  is  two,  and  the 
square  root  of  nine  is  three,  what  will  be  the  square 
root  of  four  ninths  ? 

35.  What  is  the  square  root  of  sixteen  twenty- 
fifths  ? 

36.  What  is  the  value  of  x  in  the  equation  x^  =i^^f. 

37.  WJiat  is  the  value  of  x  in  the  equation  2  x^ 

38.  What   is  the  value   of  x  in  the  equation    — 

—  a¥  • 

11 


162  INTELLECTUAL   ALGEBRA.        [(^  34 

SECTION    XXXIV. 

1.    If  X  times  x  is  z^,  and  x  times  3  is  3  a:,  whal 


Expression  will  represent  x  times  x-{-^1* 

2  If  the  expression  a;-f-4  be  multiplied  by  x,  what 
will  represent  the  product  ? 

3  If  the  expression  z-|-9  be  multiplied  by  x,  what 
will  be  the  product  ? 

4.  If  the  expression  z  -|-  3  be  multiplied  by  3,  what 
will  express  the  product  ? 

5.  When  the  expression  z  -j~  '^  ^^  multiplied  by  x, 
the  product  is  z^  -j-  3  z  ;  and  when  z  -{-  3  is  multiplied 
by  3,  the  product  is  3z-|-9-  What  will  represent 
the  sum  of  their  products  1 

6.  If  z-|-3  be  multiplied  by  z-|-3,  what  will  ex- 
press the  product  ?  that  is,  what  will  be  the  second 
power  of  z  -|-  3  ? 

First,  multiplying  z  -j-3  by  z ; 

z  times  z,  or  z  X  z  nr     z~, 
and  z  times  3,  or  z  X  3  =  3  z  ; 


then  their  sum  is    z-  -}-  3  z. 

Next,  multiplying  z  -f-  3  by  3 ; 

3  times  z,  or  3  X  2^  =  3  z, 
and  3  times  3,  or  3  X  3=  9 ; 


then  their  sum  is    3z-|-9. 
Thus,  3  z  -I-  9,  added  to  x^  +  3  z,  is 

z2-|-3z  +  3z  +  9. 
Uniting  terms,  gives  z~-}-Gz-j-9, 


which  is  the  square  or  second  power  of  z-|~'^- 

•  A  bar or  parenthesis  (  )   embraces  terms  to  be 

taken  together,  or  subi'^''*  t«  the  same  ope'aticn 


•§  34.]       INTELLECTUAL  ALGEBRA.  163 

Or   multiplying  a;  -|-  3 
by2;  +  3, 


x^-\-3  X,  X  times  x-\-S. 


Sx-\-9,  3  times  x  -\-  3. 


Sum  of  products,  a;2 -f- 6  X -|- 9,  or  square  of  x-\-3. 

It  is  evident,  from  inspecting  the  above,  that  the 
square  of  x  -f-  3  is  x^,  twice  x  X  3,  and  the  square 
of  three ;  that  is,  the  square  of  the  first  term,  and 
twice  the  first  term  multiplied  by  the  last,  and  the 
square  of  the  last  term. 

7.  What  is  the  square  root  of  the  expression  x^  -\- 
6x  +  9? 

Extracting  the  square  root  of  x~,  the  first  term  of  the 

square,  gives  x  for  the  first  term  of  the  root. 
Since  6  a;  is  twice  the  first  term  of  the  root  multiplied 
by  the  last,  that  is,  twice  x  multiplied  by  the  last 
term  of  the  root,  if  6  x  be  divided  by  twice  x,  the 
quotient,  3,  will  be  the  last  term  of  the  root. 
Therefore,  x-\-3  must  be  the  square  root  of  the  ex- 
pression x^-\-6x-\-9.     See  6th. 

8.  In  the  expression  x~-\-6x,  what  term  is  wanting 
to  make  the  expression  a  perfect  square  ? 

Extracting  the  square  root  of  x^,  the  first  term  of  the 
expression,  gives  x  for  the  first  term  of  the  root. 

Since  6  x  is  ttvice  the  product  of  the  two  terms  of  the 
root,  if  6  X  be  divided  by  2  x,  that  is,  by  ticice  the 
first  term  of  the  root,  the  quotient,  3,  will  be  the 
last  term  of  the  root. 

As  3  is  the  other  term  of  the  root,  9,  its  square,  must 
be  added  to  the  expression  t--1-0x,  to  complete 


164  INTELLECTUAL  ALGEBRA.        [§  34. 

the  square,  that  is,  to  make  the  expression  a  perfect 
square. 
Then  the  expression  x'^  -j-  G  z  -|-  9,  is  a  perfect  square 

9.  What  must  be  added  to  an  expression,  con- 
sisting of  two  terms,  only  one  of  which  is  a  second 
power,  to  complete  the  square  ? 

Extract  the  square  root  of  the  term  containing  the 
second  power,  and  divide  the  other  term  by  twice 
the  square  root  thus  found ;  the  quotient  will  be  the 
square  root  of  the  term  that  must  be  added  to  the 
expression  to  make  it  a  perfect  square.  This  is 
deduced  from  the  preceding. 

10.  To  find  the  square  root  of  a  perfect  second 
power,  consisting  of  three  terms. 

Extract  the  square  root  of  the  first  term,  which  is  a 
perfect  square,  for  the  first  term  of  the  root. 

Since  the  term,  which  is  not  a  perfect  square,  is  twice 
the  first  term  of  the  root  multiplied  by  the  last  term 
of  the  root,  divide  this  term  of  the  square  by  twice 
the  term  of  the  root  already  found,  and  the  quotient 
will  be  the  other  term  of  the  root.  This  is  evident 
from  inspection  of  the  6th  and  7th,  and  explains 
the  rule  for  extraction  in  arithmetic. 

11.  What  is  the  second  power  of  x-|-4  ? 

By  the  6th,  the  square  of  the  first  term  is  a:  X  a;  =  t-~ 

Twice  the  product  of  both  terms  is  2X2;X4  =  8  2; 

The  square  of  the  last  term  is  4  X  4  =  16. 

Adding  these  three  results,  gives  z~-\-Sx-\-  16. 

12.  What  must  be  added  to  the  expression  x^ -| 
8  r,  to  make  it  a  perfect  square  ? 


§  34.]  INTELLECTL'AL)     ALGEBRA.  165 

It  is  evident  that  the  square  of  the  second  term  of  the 
root  must  be  added. 

To  find  the  second  term  of  the  root,  divide  8  x  by 
twice  the  first  term  of  the  root,  that  is,  by  2  2;, 
which  is  twice  the  root  of  a~,  and  the  quotient,  4, 
will  be  the  second  term  of  the  root ;  therefore,  16, 
the  square  of  4,  must  be  added  to  a;^  -f"  ^  ^  to  com- 
plete  the  square. 

13.  Wliat  is  the  second  power  of  x  -\-  51 

14.  What  is  the  second  or  square  root  of  i~-\-  10  a; 
1-25? 

15.  What  must  be  added  to  the  expression  x^ -[- 
10  X,  to  make  it  a  perfect  second  power  ? 

16.  What  is  the  second  power  of  x  -\-  61 

17.  What  is  the  second  root  of  x^  -f  12  x  +  36  ? 
18     What  must  be  added  te  the  expression  x^-j- 

12  X,  to  make  it  a  perfect  square  1 

19.    What  is  the  second  power  or  square  of5-j-3? 
By  the  Gth,  the  square  of  the  first  term  is  5  X  5  rr:  25. 
Twice  the  first  term  by  the  last  is  2  X  5  X  3  =  30 
The  square  of  the  last  term  is  3  X  3  =r  9. 
Collecting  the  products,  gives  25-\-30-\-9 
)r,  multiplying     5  -f-  3 
by      5-i-3 

25  -f- 15,  or  5  times  5  -|-  3, 


15  +  9,  or  3  times  5 +  3. 


Sum  of  products,    25  -f-  30  -}-  9. 

But  25  +  30  +  9  =  64,5-|-3  =  8,and  8X8-64. 

Then    the  second    power  of  5  -j-  3    consist?    )f  the 

square  of  5  r=  25,  twice  5X3  =  30, 

and  the  square  of  3  =  9; 


106  INTELLliCTUAL     ALGEBKA.  [§  34. 

or,  the  square  of  the  first  term,  twice  the  product  of 
the  two  terms,  and  the  square  of  the  last  term  ;  as 
in  6th. 

20.  What  must  be  added  to  25  -|-  30  r^  55,  to  make 
the  sum  a  perfect  square  ? 

It  is  evident  from  the  8th  and  9th,  that  the  square  of 

the  last  term  of  the  root  must  be  added. 
To  find  the  last  term  of  the  root,  divide  twice  the 
product  of  the  two  terms,  which  is  30,  by  twice  the 
square  root  of  25,  which  is  2  X  5,  and  the  quotient, 
3,  will  be  the  last  term  of  the  root. 
Then  9,  the  square  of  3,  must  be  added  to  25 -[-30, 
giving  25  -(-  30  -j-  9  ^=  64,  a  perfect  square. 

21.  What  is  the  square  root  of  25  -f  30  -|-  9  ? 

Extract  the  square  root  of  25,  which  is  5.  Since  30 
is  twice  the  first  term  of  the  root,  multiplied  by  the 
last  term  of  the  root,  if  30  be  divided  by  twice  5, 
or  10,  the  quotient,  3,  will  be  the  last  term  of  the 
root ;  therefore,  the  root  is  5  -|-  3. 

Or,  since  25 -j- 30-J-9  rr  64,  the  square  root  of  the 
equation  is  5  -j-  3  :=  8. 

22.  What  is  the  second  power  of  10  -(-  2  ? 

23.  What  is  the  second  root  of  100  +  40  +  4  ? 

24.  What  must  be  added  to  100  -f- 40,  to  make  the 
«um  a  perfect  square? 

25.  What  is  the  square  root  of  100  -|-  100  +  25  ? 

26.  What  is  the  second  power  of  2 1  -f-  3  ? 

27.  What  is  the  square  root  of  4  «2  -j_  12  3.  _|_  9  ? 
The  square  root  of  4  2;-  is  2x;  and  the  quotient  of 


6  34.1  INTELLECTUAL     ALGEBRA.  l&lt 

12  X  divided  by  twice  2x  is  3 ;  therefore,  the 
square  root  is  2  a:  -[-  3. 

28.  What  must  be  added  to  the  expression  4  x^  -}- 
12  a;,  to  make  the  expression  a  perfect  square? 

12  a;  divided  by  twice  2  a;,  or  twice  the  first  term  of 
the  root,  is  3,  the  second  term  of  the  root ;  there- 
fore 9,  the  square  of  3,  must  be  added. 

29.  What  is  the  square  of  3  a;  -{-  4  ? 

30.  What  is  the  square  root  of  9  z^  -j.  24  3:  _|_  1(3  ? 

31.  What  must  be  added  to  9x~-|-24a;,  to  make 
the  expression  a  perfect  square  ? 

32.  What  must  be  added  to  x^-\-x,  to  make  the 
expression  a  perfect  square  ? 

It  is  evident  that  the  square  must  be  so  completed 
that  X  shall  be  twice  the  product  of  the  two  term? 
of  the  root;  then,  if  x  be  divided  by  twice  the  firs 
term  of  the  root,  the  quotient  will  be  the  seconi 
term. 

But  X  divided  by  2  x,  thus  — ,  is  equal  to  one  half; 

therefore  4  is  the  second  term  of  the  root,  and  ^ 
X  2"  — ^  ¥  ^^  *he  square  of  the  second  term  ;  there- 
fore ^  must  be  added  to  the  expression, 

and  x"  -{-  a;  -|-  i  is  a  perfect  second  power. 


33.    What  is  the  product  of  z  -j-  ^  multiplied  1  y 

Multiplying  2;  -(-  ^  by  x,  gives  x^  -\-  — . 
Multiplying  x  +  ^  by  ^  gives  j  -jr  h 

^  X 

The  products  added  are  x^-\-~ — (-  J. 


103  INTELLECTUAL  ALGEBRA.        [§  34 

Multiplying     ^-\--s 


is     z2 -| ,  or  z  times  x-\--^, 


35.    What  must  be  added  to  x~  -f-  — ,  to  complete 


2  X  1 

36.    What  is  the  square  root  of  r^ -\ 1~  IT  ^ 


and  —  -f  ^,  or  1  of  x  -f-  ^-. 

2  X  " 

Sum  of  products  is  x^-}-- — |-  i,  the  square  of  x-\-^ 

34.  What  will  express  the  square  of  x-\-  ^1 

35.  Wha 
the  square  ? 

36.  Wha 

37.  What  is  the  square  root  of  x~-{-  x-\-  ^1 

38.  What  is  the  second  power  of  x-\-^1 

39.  What  must  be  added  to  x^ -]- — ,  to  make  the 
expression  a  complete  square  ? 

X  1 

40.  What  is  the  square  root  of  x~ -\- -^  -\ 1 

41.  What  must  be  added  to  !--{-—,  to  complete 

3 

th8  square  ? 

42.  What  is  the  socona  power  of  x  -\-  ^1 

43.  What  is  the  second  root  of  x~  -f-  ~ — ^"  TT  ^ 

3  X  9 

44.  What  is  the  square  root  of  x~-\-  —  -\-  —1 

45.  What  is  the  second  power  of  x-\-}^,or  x-\-  ^1 

3  X 

46.  What  must  be  added  to  x-  -\-  —,  to  complete 

the  square  1 

47.  What  must  be  added  to  x~-\-2  r,  to  compiCte 
the  square  ? 


§  35.]  INTELLECTUAL     ALGEBRA.  169 

48.    What  is  the  second  power  o(  x  -\-  ^1 

49     Wliat  is  the  second  root  of  x~ -\ 1 ? 


SECTION    XXXV. 

1.  A  LADY  says,  "If  twice  my  son's  age  and  one 
year  more,  are  added  to  the  square  of  my  son's  age, 
the  sum.  will  be  eighty-one  years."  How  old  is  her 
son  1 

Let  2:r=the  son's  age. 
By  the  conditions  of  the  question,  x^-j-2x-|-  lnz8l 

Extracting  square  root,  x  -\-  1  =:9. 
The  square  root  of  the  first  member  is  x  -f-  1,  because 

:?  -|-  1  multiplied  by  x  -f-  I ,  is  equal  to  x-  -{-  2  x  -j-  K 

The  square  root  of  2d  member  is  nine,  because 

9  X9  =  81. 

Taking  1  from  each  member  x;=8. 

The  son's  age  is  8  years. 

2.  If  four  times  a  number  be  added  to  its  square, 
and  four  more  added  to  the  sum,  the  whole  will  be 
sixty-four.     What  is  the  number  1 

Let  X  =  the  number. 

By  the  conditions  of  the  question,  x-  -jr  4  x  -j-  4  =r  64. 

The  square  root  of  each  member  must  be  found. 

The  square  root  of  64  is  8,  because  8X8  =  64. 

Now,  the  square  root  of  the  first  term,  in  the  expression 

x~ -}- 4  X -j- 4,  is  X,  because  x  X  x=  x~. 
The  remaining  terms  of  the  expression,  that  is,  4x 


1*0  INTELLECTUAL     ALGEBRA.  [§  35. 

-j-  4,  consist  of  twice  ike  first  term  of  the  root,  that 
is,  2  2,  multiplied  by  the  last  term  of  the  root,  also 
the  square  of  the  last  term  of  the  root. 
».f  4  X,  or  twice  the  first  term  of  the  root  multiplied 
by  the  last  term  of  the  root,  be  divided  by  2  x,  or 
twice  the  first  term  of  the  root,  the  quotient  will  be 
2,  or  the  other  term  of  the  root ;  because  ^x  -\-'2 
multiplied  by  the  last  term,  2,  gives  Ax  -\-  A,  the 
remaining  part  of  the  expression  a- -|-  4  x  -|-  4  ; 

and  X  -(-  2,  or  the  square  root  of  the  expression,  mu'- 

tiplied  by  a;  -^  2,  equals  x~  -\-  A  x  -\-  A. 

Since  a  -|-  2  is  the  second  root  of  the  first  rhember,  it 

must  equal  8,  the  second  root  of  the  other  member ; 

therefore,  a:  -f  2  =  8. 

Taking  2  from  each  member,  a:  =  6,  Ans. 

8.    If  six  times  a   man's   money,  and  nine   dollars 

more,  be  added  to  the  second  power  of  his  money,  the 

sum  will  be  one  hundred  dollars.     How  much  money 

has  he  1 

4.  If  the  square  of  a  number  be  added  to  eight 
times  the  number,  the  sum  will  be  twenty.  What  is 
the  number  ? 

Let  X  z=  the  number. 
By  the  conditions  of  the  question,  x-  -{-  8  z  =  20. 
In    this    equation,   as    neither    member   is    a  perfect 
square,  the  square  of  the  last  term  of  the  root  must 
be  found  and  added  to  each  member ;   then  each 
will  be  a  perfect  second  power. 
If  8  X,  or  twice  the  product  of  both  terms  of  the  root, 
be  divided  by  2  r,  or  twice  the  first  term,  the  quo- 
tient will  be  4,  or  the  last  term  of  the  root ; 
therefore    Ifi,  the  .'square  of  t,  must  be  added  to  oaci: 


§35.]       INTELLECTUAL  ALGEBRA.  17  1 

member  of  the  equation,  to  complete  the  square  of 

each. 
Adding  IG  to  each,  x^  -f-  8  a;  -f  IG  =  20  +  16  =  3G. 
Extracting  the  square  root,  a;  -j-  4  =  6. 
Taking  4  from  each  member,  x  =  2,  Ans. 

5.  A  man  travelled  as  many  hours  as  he  travelled 
miles  in  one  hour.  If  the  whole  distance  that  he 
travelled  be  added  to  four  times  the  distance  that  he 
went  in  one  hour,  the  sum  will  be  seventy-seven  miles. 
How  many  miles  did  he  travel  in  an  hour  ?  and  what 
was  the  whole  distance  1 

G.  Complete  the  square  in  the  equation  x^-j-lGx 
=  57.     What  will  be  the  value  of  xl 

7.  What  must  be  added  to  each  member  of  the 
equation  x^  -\-  li.^  =  15,  to  make  each  a  perfect 
square  1     What  will  be  the  value  of  x1 

8.  If  the  equation  4  x^  -{-  16  x  r=  84  be  divided  by 
4,  what  will  express  the  result  ?  What  must  be  added 
to  each  member  of  the  quotient  to  complete  the 
square  ?    and  what  is  the  value  of  x  ? 

9.  What  must  be  added  to  each  member  of  the 
equation  4  x^  -f-  4  a;  =  80,  to  complete  the  square  ? 
and  what  is  the  value  of  x  ? 

The  root  of  4  x^  is- 2  x,  and  twice  2  x  is  contained  in 
4  X  once ;  therefore,  the  last  term  of  the  root  is  1, 
and  its  square,  or  1,  must  be  added  to  each  mem- 
ber, making  4  x^  -|-  4  x  -j-  1  =:  81. 

10.  What  must  be  added  to  each  member  of  the 
equation  9  x^-j-  12  x  =  21,  to  complete  the  square  1 
and  what  is  the  value  of  x  ? 

11     What  must  be  added  to  the  equation   IGx^-j- 


172         INTELLECTUAL  ALGEBRA.        [S  35 

24:  X  z=z  112,  to  make  each  member  a  perfect  seconu 
power  ?  and  what  is  the  value  of  x  ? 

12.  A  boy  has  a  number  of  pencils  in  each  hand, 
and  four  more  in  the  right  hand  than  he  has  in  the 
left.  If  the  num-ber  in  his  right  hand  be  multiplied 
by  the  number  in  his  left  hand,  the  product  will  be 
forty-five.     How  many  has  he  1 

13.  One  number  is  four  more  than  another,  and  if 
four  times  the  smaller  be  multiplied  by  the  larger, 
the  product  will  be  forty-eight.  What  are  the  num- 
bers ? 

14.  A  man,  in  buying  sheep,  gave  for  each  one  a 
number  of  dollars  equal  to  the  number  of  sheep  that 
he  bought.  If  he  had  purchased  four  times  as  many 
as  he  did,  and  eight  more,  at  the  same  rate,  they 
would  have  cost  sixty  dollars.  How  many  did  he 
buy  ?    and  at  what  price  ? 

15.  If  twenty-five  times  the  square  of  a  number  be 
added  to  fifty  times  the  number,  the  sum  will  be  two 
hundred.     What  is  the  number  ? 

10.  A  man  spent  part  of  his  money,  lost  four  dol- 
lars more  than  he  spent,  and  then  found  his  purse 
empty.  If  what  he  lost  be  multiplied  by  what  he 
spent,  the  product  will  be  ninety-six.  How  much  did 
he  spend  ?  and  how  many  dollars  had  he  at  first  ? 

17.  If  to  the  square  of  a  number  one  half  of  the 
number  be  added,  the  sum  will  be  five.  What  is  the 
number  ? 

Let  X  -^  the  number. 

By  conditions  of  the  question,  x-  -\-  -[-  r=  5. 
—  divided  by  2  x,  gives  ^,  the  2d  term  of  the  root 


^  35. j  INTELLECTLTAL     ALGEBRA.  173 


Adding  ^V,  the  square  of  i,  x'^ -{- j -{- ^\  =  5 -{-  ^^, 
or  ^-l 
Extracting  square  root,  x  -\-  ^  =i  ^ 
Reduced,  x  =z  2,  Ans. 

18.  What  must  be  added  to  each  member  of  the 

equation  x~-|-~~  =  H)  t^  make  it  a  perfect  square? 
and  what  will  be  the  value  of  a;  ? 

19.  What  must  be  added  to  each  member  of  the 

equation   x- -\ =  5^,  or  ^-^-,  to  make  it  a  perfect 

second  power  ?  and  what  is  the  value  of  a;  ? 
Completing  square,  x2  _|_  liE  _|_  ^4_  _  2^8  _|_  ^4^  _  ^4^, 

Square  root  extracted,  x  -f-  f  =:  -^^-.    Reduced,  x  =  2. 
Ans.  ^-^  must  be  added,  and  the  number  is  2. 

20.  What  must  be  added  to  each  member  of  the 

equation   x^-j =  t.  to   complete   the   square,   and 

find  the  value  of  a;  ? 

21.  In  the  equation  2;~-| ■  =  ^,  what  is  the  value 

of  X? 

22.  What  must  be  added  to 1 =^G,  to  find 

the  value  of  x  ? 

X^     .        X 

The  square  root  of  —  is  —  ;  and  twice  that  root,  or 
*  9         3 

2  X  X 

— ,  is  contained  in  — ,  the  middle  term,  one  half  of 
3  '  3  ' 

a  time ;  therefore,  4  is  the  other  term  of  the  root, 
and  I,  or  its  square,  must  be  added  to  each  mem 

'jer.  making  ^ -\- j -\- 1  =  6^  =  -V- 


174         INTELLECTUAL  ALGEBRA.       [§  36. 

Square  root  extracted,  - — |-  J-  m  |. 

Reduced,  a;  r=  6,  Ans. 

23.  A  man,  being  asked  how  much  money  he  had, 
said,  if  half  of  his  money  were  muhiplied  by  half  of 
his  money,  and  the  product  added  to  half  of  his 
money,  the  sum  would  be  thirty  dollars.  How  much 
had  he  ? 

24.  One  number  is  three  more  than  another,  and 
if  one  fourth  of  the  greater  be  multiplied  by  the  less, 
the  product  will  be  one.     What  are  the  numbers? 

25.  The  length  of  a  room  is  four  feet  more  than 
its  breadth.  If  one  half  the  length  be  multiplied  by 
half  the  breadth,  the  product  will  be  forty-eight  feet. 
What  is  the  size  of  the  room  ? 

26.  If  one  fourth  of  a  number  be  added  to  the 
square  of  a  number,  the  sum  will  be  three  eighths. 
What  is  the  number  1 

27.  A,  being  asked  vyhat  part  of  a  ship  he  owned, 
replied,  if  one  half  of  the  ship  were  added  to  one  half 
of  his  share,  and  the  sum  were  multiplied  by  one  half 
of  his  share,  the  product  would  be  three  sixteenths  of 
the  ship.     What  part  did  he  own  1 


SECTION   XXXV 1. 


1.  What  is  the  product  of  i  —  1  multiplied  by  x  ? 

2.  What  is  the  product  of  a;  —  1  multiplied  by  1  ? 

3.  If  3:2  —  2;  \^Q.  added  to  x — 1,  what  will  be  the 


pum 


§3t>.j  INTELLECTUAL     ALGEBRA.  175 

4.  If  X  —  1   be  multiplied  by  x-{-l,  what  expres 
sion  will  represent  their  product  ? 

5.  If  a; -f-  1  be  multiplied  by  x  —  1,  will  the  prod- 
uct be  the  same  as  in  the  preceding  question  T 

X  times  x-\-l  is  x^  -\-x  ;  but  this  is  once  ^-{-l  too 
many ;  therefore,  x  -\-  I  must  be  taken  from  x~-\-x; 
then  x'^-\-  X  —  x  —  1  =:  x^  —  1,  the  ?ame  as  above. 

6.  Multiply  x-{-2    by  x  —  2.     What  will    be   the 
product  ? 

The  product  will  be  x  times  x-\-'2,  less  2  times  x  -}-  9^. 

X  times  x  -|-  2  is  x^  -j-  2  x  ; 

less  2  times  x-|-2  is  —  2x  —  4. 

Sum  of  products  is  .  .  .   x-  —  4. 
Or,  multiplying  x-|-2 
by  X  — 2, 
gives  x^  -\-2x, 
and       — 2x  —  4. 
Sum  of  products  is  x-  —  4,  as  above. 
Hence,  if  only  one  of  two  factors  has  the  sign  — 
before  it,  the  product  must  have  the  same. 

7.  What  is  the  product  ofx-|-3X2:  —  3? 

8.  What  is  the  product  ofx-|-4Xa;  —  4? 

9.  What  is  the  product  of  (x  -f-  4)  X  (a;  — 5)  ? 


The  product  will  be  x  times  x  -|-  4,  less  5  times  x  -[-  4. 
X  times  x  -(-  4  is  x^  -|-  4  a; ; 
—  5  times  X -|- 4  is      — 5x  —  20. 

Sum  of  products  is  x^  —  x  —  20. 

10.  What  is  the  product  of  (x  -f  5)  X  (a:  —  7)  ? 

11.  What  is  the  product  of  (x  -f  5)  X  (x  — 4)  ? 


176  INTELLECTUAL  ALGEBRA.        f"^  36 


12.  What  is  the  product  ofx-j-GXa;  —  6? 

13.  What  is  the  product  of  x  —  6  X  x-\-5  1 

14.  What  is  the  product  of  x-fG  X  x  — 5  1 

15.  What  is  the  product  of  x-\-7  X  ^  —  7  ? 

16.  What  is  the  product  of  {z-\-7)  X  (x  — 2)  ? 

17.  What  is  the  product  of  x -f  2  X  x — 7  ] 

18.  What  is  the  product  of  x-^8  X  x  —  8 ? 

19.  What  is  the  product  of  (x  —  10)  X  (a;  -|-  10)  1 

20.  If  X  —  2  be  taken  from  x-|-2,  what  will  ex- 
j    ess  the  difference?     Vide  Sect.  XIX. 

21.  If  x4-2  X  x  +  2,  what  will  be  the  product  ? 

22.  If  x-|-2  X  X  — 2,  what  will  be  the  product  ? 

23.  If  this  last  product  {x~  —  4)  be  taken  from  the 
ni  seeding  product  (x- -}- 4  x -[-4),  what  will  express 
the  difference  ? 


4  x-f-S,  the  difference  in  the  last,  is  4  times  x-|-2, 
t^e  multiplicand.  Then  the  difference  between  the 
multipliers  is  4,  as  found  in  question  20. 


24.  What  is  the  product  of  x  —  1  X  x  ? 

25.  If  (x  —  1)  be  taken  from  (x~  —  x),  what  ex- 
pression will  represent  the  remainder  1 

It  is  evident,  if  the  whole  of  x  be  taken  from  x-  —  x, 
that  x^  —  2x,  the  remainder,  would  be  too  small  by 
one ;  because  not  the  whole  of  x  is  to  be  taken 
away,  but  x  less  1 ;  one  must  then  be  added  to  x^ 
—  2  X,  making  x^  —  2  x-j-  1. 

It  is  also  evident  that,  to  subtract  a  term,  the  sign 
before  it,  if  plus,  must  h<^  '•h"r!Jeii   vy  mnus,  and  i^ 


^  36. J  INTELLECTUAL     ALGEBRA.  17"l 

26.    What  will    express   the  product  of   x  —  1  X 
c—ll 


c  times  z  —  1  will  be  once  x  —  1  too  many  ;  therefore, 
X  —  1  must  be  taken  from  x  times  x  —  1 ;  x  —  1 
X,  X  is  x^  —  x;  less  once  x — 1  is  x  taken  away 
and  1  added;  expressed  thus,  x^  —  x  —  z-j-lr=; 
x^  —  2x-\-l. 

Thus  we  see  that  x  —  1  X  —  1,  becomes  —  x  -j-  1. 
Or,  multiplying  x  —  1 
by  a:  — 1, 

gives  x2  —  X, 
and       — x-\-l. 

Sum  of  products  is  x^  —  2x-j-l- 
Thus  it  is  apparent,  that  a  plus  term  multiplied  by  a 
plus  term  gives  a  plus  term  for  the  product,  ana  a 
minus  term  multiplied  by  a  viinus  term  gives  a 
plus  term.  A  tninus  term  by  a  plus  term,  or  a  plus 
term  by  a  minus  term,  gives  a  minus  term  for  the 
product. 

27.    What  is  the  second  power  of  x  —  1  ? 


As  it  is  X  —  1  X  ^ — 1,  it  will  be  seen,  by  inspect- 
ing the  preceding,  that  the  square  or  second  power 
of  X — 1  or  X  —  1"  is  the  square  of  the  first  term, 
I,  which  is  x^,  added  to  the  square  of  the  last  term 
—  1,  which  is  4"  1 ;  ^•''d  twice  the 

first  term,  x,  multiplied  by  the  last  term  —  1,  which 
is  —  2  X ;  connecting  terms  x^  —  2  x  -}~  1- 
28.    What  is  the  second  power  of  x  —  2,  or  (x  —  2)^? 

The  square  of  x  is  x^ ;  twice  x  X  —  2  is  — 4  x ;  and 
12 


178  INTELLECTUAL     ALGEBRA.  §  36.] 

the   square   of  — 2  is  — 2  X  — 2  =  4.     Sum  of 
products  is  x^  —  4x-j-4. 


29.  What  is  the  second  power  of  x  —  3,  or  x  —  3  t 

30.  What  is  the  second  or  squ  are  root  of  x^  —  2  x 
-j-  1,  or  /y/  x2  —  2  X  -|-  1  is  equal  to  what  ] 

This  may  be  found  by  extracting  the  root  of  the  first 
term,  and  dividing  the  middle  term  by  twice  that 
root,  because  the  middle  term  is  twice  the  product 
of  the  two  terms. 

The  square  root  of  x~  is  x ;  twice  this  root,  or  2  x,  is 
contained  in  — 2  x,  which  is  twice  the  first  term  X 
by  the  last,  — 1  time,  because  2x  multiplied    by 
— !-  1  is  —  2  X  ;  therefore,  the  root  is  x  —  1. 
J^  What  must  be   added  to  the  expression  x-  — 

2  x*o  make  the  expression  a  perfect  square  1 

32.  What  must  be  added  to  x-  —  4  x,  to  make  the 
expression  a  perfect  square  1 

33.  What  is  the  square  root  of  x-  —  4  x  -(-  4  ? 

34.  What  must  be  added  to  x^  —  6x,  to  make  the 
expression  a  perfect  square  ? 

35.  What  is  the  square  root  of  x^  —  6  x  -(-  9  ? 

36.  What  is  the  second  power  of  x  —  4? 

37.  What  must  be  added  to  x^  —  8x,  to  make  the 
expression  a  perfect  square  ? 

38.  What  is  the  second  root  of  x^  _  8  x  -f  16  ? 

39.  What  is  the  second  power  of  x  —  5  ? 

40.  What  must  be  added  to  x^ —  10  r,  to  complete 
the  square  ? 

41.  What  is  the  second  root  of  x^  —  10  x  -}-  25  ? 

42.  What  is  the  second  power  of  x  —  ^1 

The  square  of  the  first  term,  x,  is  x-;  twice  the  fitf 


[§36. 


lNT£LL.t:CTUAL     ALGEBRA. 


171^ 


term,  x,  X  by  the  last,  —  4,  is  — x,  and  the  square 
of  the  last  term  —  ^  is  +  ^.     Ans.  x~  —x-\-  }. 
Or,  multiplying  x  —  ^ 
by  X  — I, 


and      —  „  -|-i. 


Sum  of  products  is  x^  —  x  -\-  -J-,  as  above. 

43.  What  must  be  added  to  z~  —  x,  to  mak"  the 
expression  a  perfect  square  ? 

44.  What  is  the  second  root  of  x^  —  x-^-^l 

45.  What  is  the  second  power  of  x  —  ^  ? 

2  X 

46.  What  must  be  added  to  x~ ,  to  make  the 

expression  a  perfect  square  1 

2  X 

47.  What  is  the  square  root  of  x^ }-  ^  ] 

48.  What  is  the  second  power  of  x  —  |  ? 

4  X 

49.  What  must  be  added  to  x^ to  complete 

3  ' 

the  second  power  ? 

50.  What  is  the  second  root  of  x^ +  I  ' 

51.  What  is  the  second  power  of  x  —  a? 


52.  To  what  is  yX  a;2  _  if  _|_  ^4_  gquaH 

53.  To  what  is  {x  —  ^V)^  equal  ? 

54.  To  what  is  ^/a;2_^-}- -j-ig  equal  ? 

6. 


IBO 


INTEI.LKCTUAL     ALGEBRA. 


SECTION   XXXVII. 


§37.1 


1.  John  and  James  liad  equal  r.ums  of  money 
John  lost  two  cents,  and  then  the  product  of  James's 
money  multiplied  by  John's  lacked  but  one  cent  of 
bein^  a  dollar.     How  much  money  had  each  ? 

I  et  X  =  the  number  of  cents  each  had  at  first ; 

then  X  —  2  =  the  number  John  had  left. 

X    -  2  X  a"  =  2;-  —  2  x,  the  product  of  what  each  had 

left. 

By  the  conditions  of  the  question,  x^  —  2  x  -|-  1  =  100 

Extracting  the  second  root  of  each  member, 

X— 1  =  10. 

Adding  1,  xrrr  11  cents,  each  had. 

2.  One  number  is  four  less  than  another,  and  theii 
product  is  forty-five.     What  are  the  numbers  ? 

Let  X  =  the  greater  ; 
then  X  —  4  =  the  less. 
X  —  4  X  x  =  x-  —  4x,  the  product  of  the  two. 
By  the  conditions  of  the  question,  x-  —  4  x  n:  45 
The  square  of  each  member  must  be  completed,  so 
that  the  second  root  of  each  member  can  be  found. 
Twice  the  product  of  the  two  terms  of  the  root,  that 
is,  —  4  X,  must  be  divided  by  twice  the  square  root 
of  x^,  which  is  2  x,  and  this  will  give  — 2   as  the 
second  term   of  the  root ;   therefore,  the  square  of 
—  2,  which  is  -|- 4,  must  be  connected  with  pach 
member. 

Adding  4,  x^  ^  4  x  -f  4  =  40. 

Extracting  second  root,  x  —  2  =  7. 

X  =:  9,  the  greater,  ami  x  —  4  1:3  5,  the  less 


[§  37.  INTELLECTUAL     ALGEBRA.  Ibl 

3.  What  mast  be  added  to  each  member  of  the 
equation  x^  —  Gxz=:7,  to  make  each  a  perfect  second 
power  ? 

4.  What  is  tlie  second  root  of  each  member  of  the 
equation  x^  —  Gx-\-Q:=z\G1  and  what  is  the  value 
of  X? 

5.  What  must  be  added  to  each  member  of  the 
equation  a;^  —  x  =  12,  to  make  each  a  perfect  square  ? 
What  will  the  equation  become  ?  and  what  will  be 
the  value  of  x  ? 

G.  What  must  be  added  to  the  equation  4  x^  —  8x 
=  GO,  that  the  square  root  of  each  member  may  be 
obtained  ?  and  what  is  the  value  of  x? 

7.  In  the  equation  x^ ^  zz:  3,  what  is  the  value 

of  X? 

8.  In  the  equation  x~ =  --,  what  is  the  value 

'  3  3' 

of  X? 

9.  i?  each  member  of  the  equation  2  x~ r= 

1^  be  divided  by  2,  what  equation  will  repj-esent  the 
quotient  ? 

10.  If  each  member  of  the  equation  3  x^ —  18  x  == 
21  be  divided  by  3,  what  will  express  the  result?  and 
what  will  be  the  value  of  x  ? 

11.  What  must  be  added  to =  3,  to  make 

4  4  ' 

each  member  a  perfect  square  ?  and  what  will  be  the 
value  of  X  ? 

12.  Ann  had  as  many  books  as  Jane ;  but  Ann 
gave  three  of  her  books  to  Jane,  and  then  Jane's 
number  of  books  multiplied  by  Ann's  rumber,  less 


Id2  INTELLECTUAL     ALGEBRA.  §  37.] 

twice    the    sum    of    their    books,   was    twenty-three 
What  number  had  each  at  first  ? 

Let  a:rr:the  number   each  had;    then   x  —  3  =  what 
Ann  had  left ;    and  i  -j-  3  r=  what  Jane  then  had. 
X  —  3  X  a;  -|-  3  =  z2  —  9,  the  product  of  the  two; 
but  this  product,  less  twice  the  sum  of  the  books, 
that  is,  less  4  x,  is  equal  to  twenty-three  books. 
By  conditions  of  the  question,  x~  —  9  —  4zz=23, 
Adding  9  to  each  member,  z-  —  -i  x  =  32. 
Completing  the  square,  x-  —  i  x -\-  i  =z  36. 
Extracting  the  second  root,  x  —  2^6..    x^=8,  Ans. 

13.  If  twice  the  square  of  some  number  be  di- 
minished by  eight  times  the  number,  the  remainder 
will  be  ten.     What  is  the  number  ? 

14.  A  farmer  sold  six  cows,  and  said,  if  the  num- 
ber he  now  had  were  multiplied  by  the  number  he 
had  at  first,  he  would  have  five  more  than  one  half  o^ 
a  hundred.     How  many  cows  had  he  at  first  1 

15.  If  one  half  of  a  number  be  squared,  and  one 
half  of  the  same  number  be  subtracted  from  the 
square,  the  remainder  will  be  two.  What  is  the 
number  ? 

16.  Two  men  received  the  same  wages  for  a 
week's  work.  But  one  spent  two  dollars,  and  then 
the  square  of  the  sum  of  what  they  both  had  left  was 
one  hundred  and  ninety-six  dollars.  What  did  each 
receive  ? 

17.  If  from  some  number  four  be  subtracted,  and 
then  one  half  of  the  remainder  be  multiplied  by  itself, 
the  product  will  be  only  one.     What  is  the  number  ? 

18.  There  is  a  fraction  whose  denominator  is  two 


§  37. J  INTKLr.ECTUAL     ALGEBRA.  I'SS 

and  if  from  the  square  of  this  fraction  one  fourth  of 
the  fraction  be  subtracted,  the  remaiT:J'?r  will  be  one 
eighth.  What  is  the  numerator?  and  U/irtthe  frac- 
tion ? 

Let  X  =  numerator,  —  =  the  fraction. 

2 

19.  One  number  is  twice  another.  If  six  be  sub- 
tracted from  the  greater,  and  then  one  sixth  of  the 
remainder  be  multiplied  by  itself,  the  product  will  be 
four.     What  are  the  numbers  ? 

20.  A  man,  being  asked  his  age,  said,  that  his  age 
was  one  fourth  of  the  square  of  his  son's  age,  and 
that  the  difference  of  their  ages  was  twenty-four  years. 
What  was  the  age  of  each  1 

21.  A  man  changed  a  bank  note,  and  spent  one 
half  of  a  dollar ;  he  then  found  that  the  square  of  the 
money  he  had  left  was  a  quarter  of  a  dollar  more  than 
twenty  dollars.     What  was  the  value  of  the  note  ? 

22.  The  number  of  square  feet  in  a  square  room 
is  ninety-six  more  than  the  number  of  feet  in  the  sum 
of  its  sides.  What  is  the  length  of  one  side  of  the 
room  ? 

23.  If  5  be  subtracted  from  each  member  of  the 
equation  x-  —  8  2;-|-5:=14,  what  equation  will  ex- 
press the  remainder  ?  What  must  now  be  added  to 
each  member,  to  make  it  a  perfect  second  power?  and 
what  is  the  value  of  z  ? 

24.  What  is  the  value  of  a:  in  a;2  _  4  2;  _|_  7  —  103  ? 

25.  If  3  be  added  to  each  member  of  the  equation 
7^  —  2a:  —  3^=45,  what  will  the  equation  become? 
What  will  the  equation  be  when  each  member  is 
made  a  perfect  second  power  ?  and  what  will  be  tlie 
Talue  of  r  ? 


184  INTELLECTUAL  ALGEBRA.       §38.1 


SECTION   XXXVIU. 

1.  When  the  value  of  x  is~7,  x  —  4  z=  3.  If  each 
member  be  multiplied  by  itself,  what  -will  the  equa- 
tion be? 


x_4  X  x  — 4  =  i2_83:-[-lG,  and  3X3  =  9,  and 
the  equation  is  x^  —  8  x  -(-  16  :=  9. 

2.  When  the  value  of  x  is  1,  what  will  x  —  4  equal  ? 
It  IS  evident  that,  if  x  represent  7  dollars,  x  —  4  will 

equal  3  dollars,  as  above.  If  x  =  4  dollars,  then 
X  —  4  will  equal  nothing;  and  if  x^=.\,  then  x  —  4 
must  be  represented  by  —  3,  or  x  —  4  =  —  3.  For, 
if  a  man  has  only  1  dollar  in  his  purse,  and  is 
called  upon  to  pay  a  debt  of  4  dollars,  it  is  evident 
that  he  can  pay  but  1  dollar,  which  is  the  whole 
debt  less  3  dollars;  therefore  — 3  will  represent  the 
difference  or  deficiency. 

3.  When  the  value  of  x  is  1,  x  — 4  =  —  3.     If 

each  member  be  multiplied    by  itself,  what  will  the 
equation  be  ? 


z  —  4'  =  x-  —  8  X -j- 16,  as  above.  Since  a  minus 
quantity  multiplied  by  a  minus  quantity  gives  a 
plus  product,  — 3X — 3  will  be  -\-^,  and  the 
equation  will  be  x-  —  8  x  -|-  16  ^=  9,  as  above. 
Thus,  if  the  equation  x  —  4  r=  3,   and  x  —  4  =  — 

3,  each    be    squared,  the    »ame,    equation  will     be 
produced,  namely,  x-  —  8x-|-16=:9. 

4.  If    the    square    root   of    x^  —  8x-fl6r::9    be 
extracted,  what  will  the  equation  be  ? 


[^  38.  INTELLECTUAL     ALGEBRA.  1P6 


^x-  —  8x-j-lt)  =  ^  —  4;  A^  9  =  —  3,  or-|-3,  since 
either  multiplied  by  itself  will  produce  -|-9. 
Therefore,  x  —  4,  the  square  root  of  the  first 
member,  will  equal  either  plus  53  or  minus  3,  the 
root  of  the  second  member ;  and  the  equation  may 
be  expressed  thus  ;   x  —  4  =  4-3. 

5.  If  X  —  4  =r  J^  3,  what  is  the  value  of  a;  1- 

Adding  4  to  each  member,  a:  ::=  4  H-  3,  and  the  value 
of  z  is  4  plus  or  minus  3.     IF  the  root  is  -\-  3,  or 
positive,  then  a;  =  4  -{-  3,  or  7  ;   and  x  —  4  =:  3  will 
be  7  —  4  =  3,  as  in  Example  1st.     If  the  root  be 
—  3,  or  negative,  then  j  rr:  4  —  3,  or  1,  as  in  Ex- 
ample 3d.     The  value  of  x  is  either  7  or  1.     This 
may  be  verified  by  substituting  each  of  these  values 
for  x,  in  the  original  equation,  x^  —  8x  -f-  16  =  9. 
Putting  7  for  x,  gives  49  —  56  -f-  16  =  9  ;   and  put- 
ting 1  for  X,  1  — 8  -f-  16  =  9.     Here  each  value  of 
X  accords'with  the  algebraic  expression. 
Remark.  —  Hence  in  an  equation  of  the  second  degree,  the 
unknown  quantity  will  have  two  different  values,  either  of 
w'lich,  when  substituted,  will  satisfy  the   algebraic  expres- 
sion ;  while  only  one  of  them  will  generally  satisfy  the  con- 
ditions of  the  question.     How  to  determine  whether  the  root 
be  positive  or  negative,  and  what  is  the  true  value   of  the 
unknown  quantity,  may  be  seen  in  the  solution  of  the  fol- 
lowing problem. 

6.  A  man  paid  a  debt  of  four  dollars,  and  then 
found  that  the  square  of  the  money  left  in  his  purse 
was  nine  dollars.     How  many  dollars  had  he  at  first  ? 

Let  a;  =:=  his  money;  then  x  —  4=  what  he  had  left 
after  paying  the  debt;  then  the  square  of  x — 4 
must    equal    9    dollars,     x  —  4    is    x-  —  8x-\-l6. 


186  INTELLECTUAL     ALGEBRA.  ^38.) 

By  the  conditions  of  the  question, 

Extracting  square  root,  z  —  4  ;=  3,  and  x  :=  4  ±  3. 

If  the  root  of  9  be  -(-3,  x  =  'l-\-S=z7 :  therefore  he 
had  7  dollars ;  and,  after  paying  tlie  debt  of  4  dol- 
lars, he  had  7  —  4  =  3  dollars  left,  the  square  of 
which  is  9  dollars.  This  agrees  with  the  conditions 
of  the  question,  and  is  the  true  value  of  2". 

If  the  root  of  9  be  — 3,  a-  =  4  —  3,  or  I  ;  therefore, 
he  had  1  dollar  at  first,  and  if  he  had  but  1  dol- 
lar, he  could  not  pay  the  required  debt  of  4  dollars, 
and  have  the  required  sum  left.  This  last  value 
will  not  satisfy  the  conditions  of  the  question,  and 
cannot  be  the  true  value. 

7.  A  boy  bought  some  oranges.  Jf  he  had  bought 
two  less  at  the  same  rate,  the  number  of  oranges 
would  have  been  equal  to  the  price  of  one  orange, 
and  they  would  have  cost  him  tliirty-six  cents.  How 
many  did  he  buy  ?  and  what  did  each  cost  ? 

8.  If  twice  some  number  be  subtracted  from  its 
square,  the  remainder  will  be  thirty-five.  What  is 
the  number  ? 

9.  Boston  and  Providence  are  forty  miles  apart.  A 
man  starts  from  Boston,  and,  after  travelling  a  number 
of  miles,  finds  that,  if  twice  the  distance  he  has  trav- 
elled be  subtracted  from  the  square  of  that  distance, 
one  half  of  the  remainder  will  be  the  whole  distance 
from  Boston  to  Providence.  How  far  had  he  trav- 
elled ? 

10.  If  from  four  times  the  square  of  a  fraction, 
one  third  of  the  fraction  be  subtracted,  the  remainder 
will  be  ^.     What  is  the  fraction  ?    Lot  r  =  the  fraction 


[§38.  INTELLECTUAL      Al  (5EBRA.  1^1 

11.  A  and  B  received  the  same  sum  of  money  fur 
a  week's  wages.  A  spent  two  dollars,  and  B  four 
dollars;  then  the  product  of  A's  money  multii)lied  by 
B's  was  forty-eight  dollars.  How  much  money  did 
each  receive  ? 

12.  If  to  the  square  of  one  half  of  some  number 
one  third  of  the  same  number  be  added,  the  sum  will 
be  eleven.     What  is  the  number  ? 

13.  The  number  of  cents  that  A  paid  for  a  melon 
was  equal  to  half  the  number  of  melons  that  he 
bought.  B  bought  four  more  than  A,  at  the  same 
price,  and  they  cost  him  four  cents  less  than  a  dollar. 
What  was  the  price  of  a  melon  ?  and  how  many 
melons  did  each  buy  ? 

14.  If  half  of  some  number  be  added  to  the  square 
of  half  of  the  same  number,  the  sum  will  be  3J. 
What  is  the  number  ? 

15.  A  travelled  twice  as  far  as  B.  If  A  had  trav- 
elled four  miles  farther,  and  if  that  distance  were 
multiplied  by  the  number  of  miles  B  travelled,  the 
product  would  be  one  hundred  and  twenty-six  miles. 
How  many  miles  did  each  travel  1 

IG.  If  from  the  square  of  some  number  twice  the 
number  be  subtracted,  the  remainder  will  be  seven 
more  than  four  times  the  number.  What  is  the 
number  ? 

17.  What  is  the  value  of  x  in  the  equation  3  r^  — 
0  a;  — 18  =  12x4-30? 


188  INTELLECTUAL      A.LGEBRA.  ^39.  J 


SECTION    XXXIX. 

1     George  and  his  brother  have  §10.     If  George's 
money  be  multiplied  by  his  brother's,  the  product  will 
oe  $24.     How  much  money  has  each  ? 
Let  X  =z  George's ;   then  10  —  x  =  his  brother's,  and 


10  —  zXa^^  their  product.  By  the  conditions 
of  the  question,  24  rzr  10  a;  —  x^  ;  adding  x^,  and 
subtracting  24,  x^^^lOx  —  24;  subtracting  10  x, 
a;2_iOa;  — _24. 

Hence  it  is  evident  that,  if  the  signs  before  all  the 
terms  of  each  member  be  changed,  the  equation 
will  still  be  preserved.  For,  since  x~  is  24  less  than 
10  X,  if  we  try  to  take  10  x  from  x-,  24  will  be 
wanting,  as  expressed  by  —  24. 

Completing  the  square,  x~  —  10  x  -|-  25  =  25  —  24  =  1. 

Extracting  square  root,  x  —  5  =  ±  1 ,  and  x  :=:  5  ±  1. 

If  the  root  of  1  is  plus,  or  positive,  x  =  6,  or  George's, 
and  10  —  X  =  4,  or  his  brother's.  But  if  the  root 
is  minus,  or  negative,  x=r  4,  or  George's,  and  10  — 
X  rr:  G,  or  his  brother's.  Either  will  answer  the  condi- 
tions of  the  question,  as  it  does  not  specify  which 
had  the  most  money. 

2.  Divide  12  into  two  such  parts,  that  the  square 
of  the  less  will  be  equal  to  twice  the  greater.  If  x 
represents  the  less,  what  equation  will  be  formed  ? 
What  must  be  added  to  each  member,  that  only  the 
terms  containing  the  unknown  quantity  may  consti- 
tute one  member  1     What  are  the  parts  ? 

3.  The  united  ages  of  two  boys  are  15  years,  and 
the  square  of  the  age  of  the  younger  boy  is  5  years 
iircie  iha;- 1  vvice  the  ag*^  of  the  eld*^"-     How  old  is  each  ? 


L'5  39.  INTELLECTUAL     ALGEBRA.  189 

4.  Find  two  numbers  whose  sum  shall  be  20,  and 
the  square  of  one  third  of  the  greater  shall  be  double 
the  less.     What  are  the  numbers  ? 

5  The  sum  of  the  distances  that  Peter  and  John 
walked  is  8  miles,  and  the  product  of  the  distances  is 
12  miles.     What  distance  did  each  walk  ? 

6.  The  sum  of  two  numbers  is  7,  and  if  the  greater 
be  multiplied  by  the  less,  the  product  will  be  3  more 
than  tlieir  sum.     What  are  the  numbers  ? 

7.  A  has  more  money  than  B,  and  they  both  to- 
gether have  $14.  The  square  of  A's  money  is  $24 
less  than  14  times  his  money,  llow  many  dollars 
has  each  ? 

8.  What  number  is  that,  to  the  square  of  which 
if  21  be  added,  the  sum  will  be  10  times  the  number  ? 

9.  A  man  had  $10;  he  spent  a  part  of  it,  and  the 
square  of  what  he  spent  was  9  times  what  he  had 
left.     How  many  dollars  did  he  spend  ? 

10.  What  number  is  that,  to  the  square  of  which 
if  you  add  GO,  and  then  divide  the  sum  by  16,  the 
quotient  will  be  the  number  itself? 

11.  A  and  B,  together,  build  7  rods  of  wall,  and 
each,  by  agreement,  receives  as  many  dollars  per  rod 
as  the  number  of  rods  he  builds.  A  received  $2  less 
than  double  what  B  received.  How  many  rods  did 
each  build?  and  how  many  dollars  did  each  receive? 

12.  George  and  Charles,  together,  bought  10  books, 
and  each  paid  as  many  cents  for  one  of  his  books  as 
was  equal  to  the  number  of  books  he  bought.  George 
spent  2  cents  less  than  half  the  money  which  Charles 
spent.  How  many  books  did  each  buy?  and  how 
much  money  did  each  spend  ? 


I'90  INTELLECTUAL      ALGEBRA.  |  §  40 


SECTION   XL 

1.  The  sum  of  the  ages  of  John  and  William  is 
8  years,  and  the  product  of  their  ages  is  15.  IIow 
old  is  each  ? 

Let    x=:r  John's,    y^=z  William's;     then  x}j=.  their 

product.  « 

(L)  By  one  condition  of  the  question,    .  .    xy  =  15 

(2.)  By  another  condition, x-\-y^=^S 

(3.)  Subtracting  y  from  2d, x  =  8  —  y 

15 
(4.)  Dividing  1st  by  i, y  =  ~ 

15 

(5.)    Substituting  this  value  of  ?/,  in  3d,    x  :=  8 . 

(6.)  Multiplying  5th  by  a:, x^=iSx — 15. 

(7.)  Subtracting  8 z  from  6th,  .  .  .  x~  —  8xr=  —  15 

(8.)  Completing  the)    „      o      iie      ^a       t"        i 
^    '  r        fe  \x^ — 81-1-16=16  —  lojorl. 

square  of  7th,  ) 

(9.)  Extracting  2d  root  of  8th, x  —  4  =  ±1 

X  =:  4  db  1 ;  therefore,  x  z=  either  5  or  3. 

Putting  5  for  x  in  4th,  y  =z3,  and  3  for  x,  y  =^5. 

As  the  question  did  not  specify  the  elder,  either  value 

will   answer   its   conditions;    therefore,  John   is    3 

years  and  William  5 ;  or  John  5,  and  William  3. 

2.  The  sum  of  two  numbers  is  10,  and  their  prod- 
uct is  21.     What  are  the  numbers  ? 

3.  John  is  2  years  older  than  William,  and  the 
product  of  their  ages  is  15.     How  old  is  each  1 

4.  The  sum  of  two  numbers  is  20,  and  their  prod- 
uct is  96.     What  are  the  numbers  ? 


§40.  J  INTELLECTUAL      ALGEBRA.  191 

5  The  sum  of  two  numbers  is  G,  and  the  sum  of 
their  squares  is  20.     What  are  the  numbers  ? 

6.  The  sum  of  two  numbers  is  6,  and  the  differ- 
ence of  their  squares  is  12.     What  are  the  numbers  ? 

7.  The  sum  of  two  numbers  is  6,  and  their  product 
is  8.     What  are  the  numbers  ? 

8.  The  difference  of  two  numbers  is  1,  and  the 
difference  of  their  squares  is  9.  What  are  the 
numbers  ? 

9.  The  greater  of  two  numbers  divided  by  the  less, 
is  equal  to  the  less,  and  the  difference  of  their  squares 
is  72.     What  are  the  numbers  ? 

10.  A  travelled  5  miles  less  than  B,  and  the  product 
of  the  distances  both  travelled  is  84  miles.  How 
many  miles  did  each  travel  ? 

11.  What  is  that  fraction  which  will  be  equal  to 
^,  if  2  be  added  to  its  numerator,  and  if  the  numera- 
tor be  taken  from  the  denominator,  the  difference  will 
be  7? 

Let  X  =1  numerator,  w  =  denominator,  —  =z  the  frac- 

y 
tion. 

12.  There  are  two  numbers,  such  that,  if  the 
greater  be  divided  by  the  less,  the  quotient  will  be  3, 
and  their  difference  is  4.     What  are  the  numbers  ? 

13.  There  are  two  numbers  whose  sum  is  5,  and 
whose  product  is  6.     What  are  the  numbers  ? 

14.  There  are  two  numbers,  such  that,  if  ^  of  the 
greater  be  added  to  4  of  the  less,  the  sum  will  be  the 
less  number,  and  their  product  is  32.  What  are  the 
numbers  ? 

""5.  If  1  be  added  to  the  denominator  of  a  fraction, 
uie  fraction  will  be  \,  and  the  prodnc   c''  the  numer 


192  INTELLECTUAL     ALGEBRA  §  41.] 

ator    multiplied  by  the    denominator  is  (i.     What  is 
the  fraction  1 

16.  George  is  1  year  older  than  Anna,  and  the 
difference  between  the  squares  of  their  ages  is  19. 
How  old  is  each  ? 

17.  The  product  of  two  numbers  is  12,  and  theii 
difference  is  1.     What  are  the  numbers? 

18.  If  the  greater  of  two  numbers  be  multiplied 
by  the  less,  the  product  will  be  10,  and  the  difference 
between  the  two  numbers  is  3.  What  are  the  num- 
bers? 

19.  A  boy  bought  an  orange  and  3  lemons  for  11 
cents,  and  the  price  of  a  lemon  multiplied  by  the 
price  of  an  orange  was  10  cents.  What  was  the  price 
of  one  of  each  ? 


SECTION   XLI. 

1.  If  t  be  multiplied  by  x,  the  product  is  x^,  the 
second  power  or  square  of  x.  If  x~  be  multiplied  by 
X,  the  product  will  be  a:-',  that  is,  the  third  power  or 
cube  of  x ;  and  the  third  root  or  cube  root  of  x^ 
must  be  x. 

2.  What  is  the  product  of  a;  X  a;  X  a;  ? 

3.  What  is  the  product  of  3  X  3  X  3  ?  or  what  13 
the  cube  of  3  ? 

4.  What  is  the  third  or  cube  root  of  27  1 

5.  W^hat  is  the  cube  or  third  power  of  2  ? 

6.  What  is  the  cube  or  third  {Km-er  of  4  ? 

7.  What  is  the  cube  root  of  8  ? 


fs^ill  INTELLECTUAL     ALGEBRA,  IJJS 

8.  What  is  the  third  root  of  (34  ? 

9.  What  is  the  cube  of  2  x  ? 

1 0.  What  is  the  cube  of  4  x  ? 

11.  What  is  the  third  power  of  5  x  ? 

12.  What  is  the  third  root  of  64  x^  1 

13  What  is  the  cube  root  of  8  xS  ? 

14  What  is  the  cube  root  of  125x3  ? 

15  If  x2  be  divided  by  x,  the  quotient  will  be  x. 
If  x3  be  divided  by  x,  what  will  the  quotient  be  ? 

16.  Divide  x^  by  z^,  what  will  the  quotient  be  1 

17.  Divide  64  x^  by  4  x,  what  will  be  the  quotient  1 

18.  Divide  125x3  by  25  x^,  what  will  be  the  quo- 
tient ? 

19.  Extract  the  cube  root  of  8  x^  ? 

20.  Extract  the  third  root  of  27  x^  ? 

21.  The    product   of   —  X  ^   is   — ,  and  —  X  ~ 

^  2  2  4  '  4  2 

is  — .     What  is  the  cube  root  of  ^— ? 
8  8 

22.  What  is  the  product  of  -  X  -  X  —? 

23.  What  is  the  third  power  o''  —  ? 

24.  What  is  the  third  root  of  —  ? 

8 

8  x^ 

25.  What  is  the  cube  root  of  —  t 

64 

26.  What  is  the  third  power  of  —  ? 

27.  What  is  the  cube^of  —  ? 

4 

28.  What  is  the  cube  root  of —?  of—? 

64  ,      64 

13 


194  INTELLKCTUAL     ALGEBRA.  [§^'* 

29.  What  is  tlie  third  power  of  —  ?  of  —  ?  of  —  ? 

'  0  5  6 

30.  What  is  the  cube  root  of  —  ?  of  —  ? 

125  125 

.    .        27  x3         9  a2 

31.  Divide  ^^— 17  by  -^,  what  will  the  quotient  be? 

32.  If  X  =  2,  to  what  will  the  cube  of  x  be  equal  ? 

33.  If  X  r=:  3,  what  will  x^  equal  ? 

34.  If  x^  z=z  8,  what  will  x  equal  ? 

35.  What  is  the  cube  root  of  the  equation  a:^  rr  27  ? 

36.  If  a:  r=  4   be  raised  to  the  third  power,  what 
will  the  equation  be  ? 

37.  What  is  the  cube  root  of  x^  =z  125  ? 

38.  In  the  equation  a-^  r=:  64,  what  will  be  the  value 
of  I? 

39.  Wliat  is  the  third  power  of  4=  3  ? 

40.  What  is  the  cube  of  the  eciuation  ^^r=  4  ? 

'  3 

X  3 

41.  What  is  the  cube  root  of  the  equation  —  nz  27  ? 

42.  In   the   equation   —  =:  64,  what  is  the  value 

of  X  ? 

27  .t3 

43.  In  the  equation    '^—  z=i  64,  what  is  the  value 

Gi 

of  Xl 

44.  In  the  equation   8  x-'  =  64,  what  is  the  value 
of  z? 

45.  Li  the  equation  27  x^  =  27,  what  is  the  value 
of  a;? 

46.  If  the  equation  x^izz  ^x  be  divided  by  r,  what 
will  be  the  result  ?     What  will  be  the  value  of  x  ? 

47.  If  x^  be  multiplied   by  x^,  the  product   is  x^. 
Wliat  will   be  the  product  of  2  3-  uuiUiplicd  by  2x-l 


(^  42.  1  INTELLECTUAL     ALGEBRA.  195 

48.  What  is  the  fourth  power  of  .t,  or  xY,%y  % 

49.  What  is  the  square  root  of  x'^ '  of  16  a;'*  ? 

50.  What  is  the  fourth  root  of  x^  %  of  16 a;*? 

51.  What   is  the   fourth  power  of  3a;?    of  2r? 

—  ?  of  -  ?       - 

2  3 

52.  What  is  the  fourth  root  of  81  rc^?  of  —  ? 

16 


SECTION   XLII. 

1  A  MAN,  being  asked  the  age  of  his  son,  said, 
'  If  »le  square  of  his  age  be  multiplied  by  his  age, 
the  i>j-o(tjct  will  be  81  times  his  age."  What  was  the 
son's  age  ? 

Let  X  =  his  age  ; 

then  X-  X  a:  =  x-,  the  square  of  his  age. 

By  the  conditions  of  the  question,  x^  X  ^=^81  x,  or 

x3  — 81x. 

Dividing  by  x,  x^  =  81. 

Extracting  square  root,  x  =  9,  Ans. 

2.  If  x3  r=  81  X,  what  does  x^  equal  ?  What  does 
X  equal  ? 

3.  The  cube  of  a  number  is  27.  What  is  the 
number  ? 

Let  X  =i  the  number ;  then  x^  =  27. 
Extracting  the  cube  root  of  each  member,  x  =i  3. 

4.  If  X  be  multiplied  twice  by  x,  it  will  be  expressed 
thus;  X  X  a;  X  a;  =  x3.  What  will  express  the  product 
of  2  X  multiplied  by  itself  twice  ? 


)  96  INTEJuLECTUAL     ALGEBRA.  [§  42. 

5.  If  a  boy's  money  be  taken  from  the  cube  of  his 
money,  the  remainder  will  be  15  times  his  money. 
How  many  dollars  has  he  ? 

G.  The  cube  of  a  number  is  4  times  the  number 
What  is  the  number  2 

7.  The  cube  of  a  number  15  16  times  the  square 
of  the  same  number.     What  is  the  number  ? 

8.  The  cube  of  a  number  is  64.  ^V'hat  is  the 
number  ? 

9.  What  must  be  the  side  of  a  cubical  box  to  con- 
tain 125  cubic  feet  ? 

10.  The  cube  of  one  half  of  a  number  is  twice  the 
number  itself     What  is  the  number? 

11.  If  the  second  power  of  a  number  be  multiplied 
by  ^  of  the  number,  the  product  will  be  16.  W^hat 
is  the  number  ? 

12.  A's  age  is  the  square  of  B's,  and  C's  is  the 
product  of  A's  multiplied  by  B's,  and  the  sum  of 
their  ages  is  21  times  B's  age.  What  is  the  age  of 
each  ? 

13.  If  from  the  third  power  of  a  number  4  "times 
the  second  power  of  the  same  number  be  subtracted, 
the  remainder  will  be  4  times  the  square  of  the 
number.     What  is  the  number  ? 

14.  If  from  the  cube  of  a  number  the  square  be 
subtracted,  the  remainder  will  be  6  times  the  number. 
What  is  the  number  ? 

15.  If  from  the  cube  of  some  number  60  be  sub- 
tracted, only  4  will  remain.     What  is  the  number  1 

16.  What  must  be  the  side  of  a  cubical  box  con- 
taining 216  cubic  feet  ? 

17     A  man  said,  if  10  times  his  money  were  taken 


[§  43.  INTBXLECTUAL     ALGEBRA  1§/ 

from  the  cube  of  his  money,  the  remainder  would  be 
9  times  the  square  of  his  money.  How  many  dollars 
had  he  ? 

18.  The  cube  of  a  number,  less  2.">,  is  100.  What 
is  the  number  ? 

19.  A  man,  being  asked  the  age  of  his  son,  said, 
"  If  3  times  the  square  of  his  age  be  taken  from  the 
fourth  power  of  his  age,  the  remainder  will  be  6  times 
the  square  of  his  age."      How  old  was  his  son  ? 

20.  A's  money  is  the  cube  of  B's,  and  if  20  times 
B's  be  taken  from  A's,  the  remainder  will  equal  the 
square  of  B's.     How  many  dollars  has  each? 


SECTION    XLIII. 

1.  Since  I  r=  2,  and  f  ==  2,  therefore  I  —  6  jjere 
2  has  the  same  relation  or  ratio  to  4,  that  3  has  to  6. 
This  relation  may  be  expressed  thus'; 

2  :  4  =  3  :  G; 

that  is,  the  ratio  of  2  to  4  equals,  or  is  the  same  as, 

the  ratio  of  3  to  0. 

2.  In  the  above  equation  of  ratios,  4  is  the  same 
part  of  2  that  G  is  of  3,  and  2  is  the  same  part  of  4 
that  3  is  of  G. 

3.  From  the  above  proportion,  or  equality  of  ratios, 
it  is  apparent  that  the-  product  of  the  extremes  that 
is,  2  X  G,  is  equal  to  the  product  of  the  means,  that 
is,  4  X  3 ;  or  2  X  G  =  4  X  3,  since  these  products  are 
the  same. 


198  INTELLECTUAL.     ALGEBRA.  §  43."] 

4.  To  what  number  has  3  the  same  ratio  or  rela- 
tion that  2  has  to  4  ? 

Let  X  =  the  number  ;  then  2  :  4  :=  3  :  a:.  Multiply- 
ing extremes  and  means,  as  in  3d,  2  z  r=  12,  and 
a;r=  6,  Ans. 

5.  What  number  has  the  same  ratio  to  G  as  2  to  4  ? 
c  =z  number  ;  then  2:4  =  a;:6;    4X*;=;2XC,  or 

4  a;  zr:  12  ;     z  :=  3,  Ans. 

6.  To  what  number  has  2  the  same  ratio  as  3  to  6? 
T  =:  number  ;  then  2:2-:=3:6;    3X2;=:2XG,  or 

3  z  =  12  ;    z  =  4,  Ans. 

7.  What  number  has  the  same  ratio  to  4  as  3  to  6  ? 
X  =  number  ;  then  z:4rr3:6,-    CX2;:=4X3,  or 

6  X  =  12  ;  z  =:  2,  Ans. 
S.  Thus,  if  any  three  terms  in  an  equality  of  ratios 
be  known,  the  other  may  be  found ;  that  is,  dividing 
the  product  of  the  means  by  one  extreme,  gives  the 
other  extreme,  and  dividing  the  product  of  the  ex- 
tremes by  either  one  of.  the  means,  gives  the  other. 
Hence  the  "  Rule  of  Three,"  or  "  Proportion." 

9.  A  man  gave  $8  for  4  sheep.    What  will  5  sheep 
cost  at  the  same  rate  ? 

Let  X  =z  cost  of  5  sheep  ;  then  4  :  8  ^=  5  :  z.     By  the 
3d,  4  X  :c  =  o  X8,  or  4z:=:40.    z  =  10.    ^ns.  $10. 

10.  If  2  oranges  cost  8  cents,  what  will  7  oranges 
cost  ? 

11.  If  4  writing-books  cost  24  cents,  what  will  3 
cost? 

12.  If  2  cows  cost  840,  what  will  5  cows  cost? 


5  43.]  l>fTELI,FXT[:AL     ^ALGEBRA.  1S9 

13.  Two  numbers  are  to  each  other  as  3  to  4,  and 
their  product  is  48.     What  are  the  numbers  ? 

Let    X  =  less,  and   y  =:  greater  ;    then   z  :  y  =  3  :  4, 
andzy  =  48.    By  3d,  4  X  a:  =  3  X  y,  or  4  z  =  3y. 

z  =  — .     Put  —  for  z,  — ^  X  1/  =  4*^,  or  -i-  z=  48. 

i  4  4  -^  4 

^=16.     y2  =  64.     y:=S.     Put  8  for  y,  z  = -^^i 
=  6.  Ans.  6  and  8.  . 

14.  If  z  is  to  y  as  2  to  -5,  what  part  of  y  is  z  ? 

2  y 

z  :  y  =^  2  :  .5.   Bv  3d,  .5  z  =  2  y.   x  =i  '^-,  or  f  of  y,  ^4«5. 

5 

15.  If  one  number  is  to  another  as  3  is  to  7,  and 
if  X  represents  the  greater,  what  part  of  z  will  repre- 
sent the  less  ? 

Let  y  =  the  less  ;  then  y  :  z  r=  3  :  7.    By  3d,  7  y  m  3  z. 

.3  z  3  X      . 

y  ^  -^7,  and  -^  will  represent  the  less  number. 

7  7 

16.  Two  numbers  are  to  each  other  as  3  to  4,  and 
their  difference  is  3.     What  are  the  numbers  ? 

17.  Two  numbers  are  to  each  other  as  1  to  3,  and 
the  square  of  their  sum  is  64.    What  are  tJie  numbers  ? 

Let  y  :=  smaller,  and  z  =  larger;  then  y  :  z  r=:  1  :  3. 

By  3d,  3  y  =:  z.     y  ==  — - ;  then   z  zn  larger,  and  — 

,        X  i  X 

==.  smaller,     z  -j =  — ,  their  sum.     By  the  ques- 

16  X*  4x        _  X 

tion,   =^64.      — :=  ■;.      z  rr  6.     —  =  "2. 

9  3  3 

Ans.  2  and  6. 
IS.    Two  numbers  are  to  each  other  as  2  to  3,  and 


200  INTELLECTHAL  ALGEBRA.        [§  43 

the  difference  of  their  squares  is  20.      What  are  the 
numbers  ? 

19.  Two  numbers  arc  to  each  other  as  4  to  7.     1 
the  less  be  subtracted  from  the  greater,  the  remaiiuler 
will  be  G.     What  are  tlie  numbers  ? 

20.  John's  money  was  to  William's  as  1  to  3 
William  spent  10  cents,  and  then  John's  was  to  Wil- 
liam's as  1  to  2.  How  much  money  had  each  at 
first? 

21.  The  difference  of  two  numbers  is  to  their  sum 
as  3  to  13,  and  their  product  is  40.  Wh  U  are  the 
numbers  ? 

22.  A's  number  of  horses  multiplied  by  B's  would 
be  15.  A  sold  one  horse  to  B,  and  then  A's  horses 
were  to  B's  as  1  to  3.  How  many  horses  had  '^ach 
at  first  ? 

23.  The  product  of  two  numbers  is  8,  and  their 
squares  are  to  each  other  as  1  to  4.  What  are  thft 
numbers  1 


INTELLECTUAL      ALGEBRA.  201 


MISCELLANEOUS   QUESTIONS. 

1.  Anna  is  3  times  as  old  as  Charles,  and  the  sum 
of  their  ages  is  12  years.     What  is  the  age  of  each  ? 

2.  What  number  must  be  added  lo  5  times  itself, 
that  the  product  may  be  54  ? 

3.  If  a  number  be  added  to  4  of  itself,  the  sum 
will  be  27.     What  is  the  number  ? 

4.  If  a  number  be  increased  by  f  of  itself,  the  sum 
will  be  21.     What  is  the  number  ? 

5.  One  number  is  6  more  than  another,  and  their 
sum  is  28.     What  are  the  numbers  .' 

0.  One  number  is  7  less  than  another,  and  their 
sum  is  23.     What  are  the  numbers  ? 

7.  The  sum  of  2  numbers  is  33,  and  their  differ- 
ence is  9.     What  are  the  numbers  ? 

8.  Daniel  lost  f  of  his  money,  and  had  12  cents 
left.     How  many  cents  had  he  at  first  ? 

9.  Levi  says,  the  diflerence  between  f  and  f  of  his 
age  is  7  years.     How  old  is  he  I 

10.  A  man  paid  away  f  of  Ins  money,  and  lost  $4. 
He  still  had  a  dollar  left.     How  many  had  he  at  first  ? 

1 1.  John  says,  "  1  have  f  of  my  books  left ;  and  if 
you  give  me  10  more,  I  shall  have  my  original  number 
complete."     How  many  had  he  at  first  ? 

12.  A  farmer  says,  "  If  I  had  as  many  more  sheep 
as  I  now  have,  ^  as  many  more,  and  ^  as  many  more, 
I  should  still  lack  7  of  having  a  hundred."  How 
many  has  he  ? 

13.  Frederic  says,  if  you  will    give    him  5  mare 


2028  INTELLECTUAL     AI.OEUKA. 

apples,  he  can  divide  wliat  he  will  then  have  among 
his  3  companions,  and  they  will  get  7  apples  apiece. 
How  many  apples  has  he  ? 

14.  In  an  orchard  of  120  trees  there  are  twice  as 
many  pear-trees  as  peach-tr.ees,  and  3  times  as  many 
apple-trees  as  there  are  of  both  the  other  kinds. 
How  many  trees  are  there  of  each  kind  1 

15.  A  man  paid  f  of  his  money  to  one  person,  and 
^  of  it  to  another,  and  still  had  $28  left.  How 
many  dollars  had  he  at  first  1 

16.  A  boy  gave  away  ^  of  his  money,  and  spent  ^ 
of  it.  He  then  had  20  cents  left.  How  much  money 
had  he  at  first  ? 

17.  Says  John  to  Samuel,  "  My  age  is  now  only  ^ 
of  yours;  but  if  we  live  4  years  longer,  mine  will  be 
I  of  yours."     What  is  the  age  of  each  ? 

18.  A  man  on  horseback  travelled  a  certain  dis- 
tance in  15  hours.  A  locomotive,  going  at  the  rate 
of  20  miles  an  hour,  travelled  tlie  same  distance  in 
3  hours.  How  many  miles  an  hour  did  the  man  on 
horseback  travel  ? 

19.  A  traveller  started  from  Boston  for  Albany  l) 
hours  before  the  cars,  and  the  train,  going  at  tlie  rate 
of  18  miles  an  hour,  overtook  him  in  4  hours.  How 
many  miles  did  he  travel  in  an  hour  ? 

20.  A  says  to  B,  "  Give  me  ^  of  your  money,  and 
I  can  spend  $2,  and  still  have  remaining  double  what 
you  would  have  left."  "  How  is  that  ?  "  says  B  ;  "  for 
1  have  f  as  many  as  you  now."  How  much  money 
hrL"  each  ? 

2'  Two  men  started,  at  G  o'clock  in  the  morning, 
one  from  Philadelphia,  and  the  other  from  New  York, 


INTELLECTUAL     ALGEBRA.  203 

90  miles  apart,  to  meet  each  other.  A  travelled  4, 
and  B  5  miles  an  hour.  At  what  time  did  they  meet"? 
and  how  far  did  each  travel  ? 

22.  Divide  17  into  2  such  parts,  that  J  of  the  one 
Bhall  be  equal  to  §  of  the  other.     What  are  t'le  parts? 

23.  A  man,  travelling  5  miles  an  hour  has  10 
hours  the  start  of  a  train  of  cars,  going  at  the  rate 
of  5  miles  to  the  man's  1.  In  how  many  hours  will 
the  train  overtake  the  man  ?  and  how  far  must  it  go 
to  do  so  ? 

24.  Five  years  ago,  Kate  was  twice  as  old  as  Abby. 
Now  Abby's  age  is  to  Kate's  as  2  to  3.  What  are 
their  ages  ? 

25.  A  receives  §  as  much  money  as  K.  After  A 
had  spent  $2,  B  had  double  what  A  had  left.  How 
many  dollars  had  each  at  first  ? 

26.  A  revenue  cutter,  in  pursuit  of  a  merchant 
ship,  sails  3  miles  to  the  ship's  2;  but  the  ship  goes  at 
the  rate  of  6  miles  an  hour,  and  has  3  hours  the  start 
of  the  cutter.  In  how  many  hours  will  the  ship  be 
overtaken?  and  how  many  miles  must  the  cutter  sail 
to  do  it? 

27.  If  5  be  added  to  5  times  a  rfumber,  ^  of  the 
sum  will  be  1  less  than  the  number.  What  is  the 
number? 

28.  A  can  plant  ^  of  a  field  in  a  day,  and  B  can 
plant  ^  of  it  in  the  same  time.  If  they  work  together, 
how  long  will  it  take  them  to  plant  it  ? 

29.  Says    Samuel    to   William,    "  Give    me    1    of 
your  apples,  and  my  number  will  be  double  of  yours. 
William  replies,  "  Give  me  1  of  yours,  and  we  shall 
each  have  the  same  Humber."     How  many  has  each'' 


204  1NTELLECT\  ALGEBRA. 

30  A  boy  wished  to  buy  a  certain  number  of  pen- 
cils, at  4  cents  apiece,  but  lacked  3  cents  of  being 
able  to  pay  for  them ;  so  he  bought  the  same  number, 
at  3  cents  apiece,  and  had  just  money  enough  left  to 
buy  one  more  at  the  latter  price.  How  much  money 
had  he  ?  and  how  many  pencils  did  he  buy  ? 

31.  What  o'clock  is  it  when  tlie  minute-hand  and 
hour-hand  are  together  for  the  fourlli  time  since  12 
o'clock  ? 

32.  A  boy  has  some  money  in  each  hand,  and  84 
in  his  purse.  When  he  takes  the  purse  in  his  right 
hand,  the  money  in  that  hand  is  double  the  money 
in  his  left  hand;  but  when  the  purse  is  in  his  left 
hand,  the  money  in  the  left  is  $2  more  than  there 
is  in  the  right  hand.  IIow  many  dollars  has  he  in 
each  hand  ? 

33.  John  bought  5  peaches  and  3  pears  for  21 
cents.  Andrew,  with  only  ^  as  much  money,  bought, 
at  the  same  rate,  2  pears  and  1  peach.  How  much 
did  they  pay  for  1  of  each  kind  of  fruit  ? 

34.  Three  times  Eliza's  age  added  to  twice  Abby's 
age  is  27  years.  If  three  times  Abby's  age  be  taken 
from  twice  Eli?a's,  the  difference  will  be  5  years. 
What  is  the  age  of  each  ? 

35.  A  steamboat,  in  pursuit  of  a  ship,  sails  3  miles 
while  the  ship  sails  2;  but  the  ship  started  5  hours 
before  tlie  steamboat,  and  averages  8  miles  an  hour. 
How  many  miles  must  the  steamboat  go  to  overtake 
the  ship  ?  and  how  many  hours  will  it  take  to  do  it  ? 

36.  A  farmer  has  his  cows  in  2  pastures,  and  one 
pasture  has  in  it  f  as  -many  as  the  other.  He  took  1 
cow  out  of  the  pasture  containing  the  less  number 


INTELLECTUAL     ALGEBRA.  205 

and  put  her  in  the  other ;  and  then  the  latter  con- 
tained double  the  number  in  the  former.  How  many 
cows  were  in  each  pasture  at  first  ? 

37.  The  quotient  of  one  number  divided  by  another 
is  3j  and  their  difference  is  4.      What  are  the  numbers? 

3S.  The  length  of  a  room  is  9  feet  more  than  the 
breadth,  and  the  number  of  square  feet  in  it  equals 
10  times  the  length  of  the  room.  What  is  the  length 
of  the  room  1 

39.  The  difference  between  two  numbers  is  3,  and 
their  product  is  28.     What  are  the  numbers  ? 

40  A  man  bought  3  calves  and  4  sheep  for  $27. 
He  afterwards,  at  the  same  rate  at  which  he  pur- 
chased, returned  2  calves  and  I  sheep  to  the  seller, 
and  received  back  <^13.  What  was  the  price  of  1 
of  each  ? 

41.  If  I  of  a  number  be  multiplied  by  ^  of  the 
same  number,  the  product  will  be  72.  What  is  the 
number  ? 

42.  A  farmer  said,  if  ^  his  number  of  cows  were 
multiplied  by  ^  of  the  number,  the  product  would  be 
his  number  of  cows.     How  many  had  he  ? 

43.  If  i  be  added  to  the  quotient  of  10  divided  by 
some  number,  the  sum  will  be  3  times  that  number  I 
What  is  the  number  ? 

44.  If  |-  of  a  number  be  multiplied  by  i-  of  the 
same  number,  and  from  the  product  J  of  the  number 
be  taken,  the  remainder  will  be  2.  What  is  the 
number  ? 

45.  The  sum  of  two  squares  is  100,  and  their  differ- 
ence is  28.     What  are  their  square  roots  ? 

46.  Divide  30  into  two  such  parts,  that  the  greater 


20t)  INTELLECTUAL     ALGEBRA 

divided  by  the  square  of  the  k.ss,  shall   be  equal  to  3 
What  are  me  numbers  ? 

47.  Matilda  had  as  much  money  as  Catharine, 
but  Catharine  gave  5>3  to  Matilda,  and  then  Matilda's 
money,  multiplied  by  Catharine's,  was  $40.  How 
many  dollars  had  each  1 

48.  One  number  is  to  another  as  1  is  to  2,  and 
their  product,  less  the  smaller  number,  is  G.  What 
are  the  numbers  ? 

49.  If  John's  money  be  taken  from  3  times  Hen- 
ry's, and  ^  of  the  remainder  be  added  to  ^  of  the  dif- 
ference between  Henry's  and  twice  John's,  the  sum 
will  be  $14  ;  but  half  of  John's  money  is  $2  more 
than  ^  of  Henry's.     How  many  dollars  has  each? 

50.  The  difference  between  the  numerator  and  de- 
nominator of  a  proper  fraction  is  6;  and  if  2  be  taken 
from  the  numerator,  and  added  to  the  denominator, 
the  fraction  will  be  ^.     What  is  the  fraction  ? 

51.  What  o'clock  is  it  when  the  square  of  the  time 
past  from  midnigJit  is  equal  to  the  remaining  time  to 
noon  ? 

52.  If  the  denominator  of  a  fraction  be  divided  by 
the  numerator,  the  quotient  will  be  4  ;  and  if  the  nu- 
merator be  multiplied  by  the  denominator,  the  product 
will  be  4.     What  is  the  fraction  1 

53.  One  number  is  the  square  of  another,  and  if 
the  less  be  increased  by  2,  and  the  sum  multiplied  by 
the  greater,  the  product  will  be  24  times  the  smaller. 
What  are  the  numbers  ? 

54.  A  man  started  from  Boston  to  go  to  Hartford, 
a  distance  of  100  miles.  After  travelling  a  short 
time,  he  found,  if  2i  times  the  distance  he  had  trav 


INTELLECTUAL     ALGEBRA.  207 

elled  were  taken  from  the  square  of  half  that  distance, 
the  remainder  would  be  f  of  the  whole  distance  to 
Hartford  added  to  half  the  distance  he  had  travelled. 
How  far  had  he  travelled  ? 

55.  The  difference  of  two  numbers,  multiplied  by 
the  less,  is  twice  the  less,  and  twice  the  greater  added 
to  the  less  is  16.     What  are  the  numbers  1 

50.  A  had  2  dollars  to  B's  3 ;  and  after  counting 
their  money,  they  found  that,  if  2  dollars  were  taken 
from  half  of  A's  money,  and  the  remainder  were  mul- 
tiplied by  f  of  B's,  the  product  would  be  double  B's 
money.     How  many  dollars  had  each  ? 

57.  A  room  contains  120  square  feet,  and  the  dif- 
ference between  the  length  and  the  width  is  2  feet. 
How  long  and  how  wide  is  the  room  ? 

58  Th(5  product  of  two  numbers  is  50,  and  their 
quotient  is  2.     What  are  the  numbers  1 

59.  The  length  of  a  fence  is  10  times  its  height, 
and  the  number  of  square  feet  in  the  fence  is  equal 
to  twice  the  cube  of  the  height.  How  long  and  how 
high  is  the  fence  ? 

60.  The  product  of  Sarah's  money,  multiplied  by 
|-  of  Eliza's,  is  equal  to  Sarah's,  and  the  square  of 
Eliza's  is  $20  less  than  the  square  of  Sarah's.  How 
many  dollars  has  each  ? 

61.  On  asking  the  number  of  cannon  balls  'n  a 
certain  pile  at  the  Navy  Yard,  an  officer  replied,  "  If 
I'j  of  the  number  of  balls  be  multiplied  by  -^  of  the 
number,  and  from  the  product  f  of  the  number  be 
taken,  the  remainder  will  be  20  balls."  How  many 
balls  were  there  in  the  pile  ? 

62.  Two  men  left  Philadelphia  for  Baltimore,  a  dis- 


208  INTELLECTUAL      ALGEBRA 

tance  of  100  miles,  at  5  o'clock,  A.  M.  A  travelled 
2  miles  an  hour  faster  than  B,  and  B  was  2J-  hours 
longer  on  the  way.  How  fast  did  each  travel  ?  and 
at  what  o'clock  di  1  each  reach  Baltimore? 

63.  Divide  ]4  into  2  such  parts,  that  if  the  product 
of  the  parts  be  divided  by  the  second  power  of  the 
smaller  part,  the  quotient  will  be  to  the  greater  part 
as  1  is  to  4.     What  are  the  parts  ? 

64.  The  difference  between  the  length  and  breadth 
of  a  room  is  5  feet.  If  9  feet  be  taken  from  §  of  the 
number  of  square  feet  in  the  room,  the  remainder  will 
be  the  number  of  square  feet  in  a  squ«are  room,  whose 
side  is  1  foot  less  than  the  breadth  of  the  room  whose 
dimensions  are  required'.  How  long  and  how  wide  is 
the  room  ? 

65.  There  is  a  bridge  100  rods  long,  and  the 
square  of  one  fourth  of  tlie  distance  from  the  north 
end  of  the  bridge  to  the  middle  of  the  draw,  less  once 
that  distance,  is  equal  to  the  distance  from  the  middle 
of  the  draw  to  the  other  end  of  the  bridge.  How  far 
is  the  middle  of  the  draw  from  each  end  of  the  bridge  l 

66.  If  A's  money,  which  is  $12,  be  divided  by  B's 
money  less  $2,  the  quotient  will  be  81  less  than  B'a 
money.     How  many  dollars  has  B  ? 

67.  Peter  is  4  years  older  than  John,  and  half  tho 
product  of  their  ages,  added  to  the  sum  of  their  ages 
is  equal  to  the  sum  of  Peter's  age,  added  to  the 
square  of  John's  age.     What   are  their  ages  ? 

68.  What  o'clock  is  it  when  the  time  past  from 
noon  to  midnight,  taken  from  the  square  of  the  time 
past,  is  equal  to  the  time  from  noon  to  midnight  ? 


KccoinmmbationB  anh  Noticea 

OF 

TOWEirS  INTELLECTUAL  ALGEBKA. 


The  subscribers,  Principals  in  the  Department  of  Mathematics 
in  the  Public  Schools  of  Boston,  have  examined  D.  B.  Tower'3 
■'  Intellectual  Jllgchra"  and  are  well  pleased  with  the  Work.  They 
believe  that  the  careful  and  minute  analysis  of  questions  in  it  is 
calculated  to  train  the  mind  of  the  pupil  to  correct  habits  of  inves- 
tigation, and  they  cordially  recommend  it  to  the  consideration  of 
iflose  interested  in  education. 

Peter  JVlACiiiNTosn,  Jk.  James  Robinson, 

Levi  Cona.\t,  Aaron  D.  Capen, 

JosiAii  Fairbank,  Nathan  Merrill, 

Reuben  Swan,  Jr.  John  A.  Harris, 

LoRiNG  Lathrop,  Charles  Kimball, 

Joseph  Hale,  William  A.  Shephard 

Jonathan  Battles,  Jr.  BE.NjAMiN  Drew,  Jr. 

June  28th,  1845. 

Boston,  June  30th,  1845. 
We  have  examined  the  "  Intellectual  Algebra"  by  D.  B.  Tower,  and 
we  are  glad  to  find  that  the  hitherto  perplexing  science  of  Algebra 
is  so  simplified  and  so  clearly  illustrated,  as  to  render  it  easily  at 
tuinable  by  the  younger  classes  of  children. 

Mr.  Tower  has  the  merit  of  originality  in  his  conception  of  an 
"  Intellectual  Algebra."  The  value  of  this  work  is  much  enhanced, 
not  merely  from  the  fact  that  the  author  ranks  high  as  a  Mathema- 
tician;  but  in  an  especial  manner,  since  he  has  been  a  successful 
Teacher  in  this  department,  and  is  thoroughly  vereed  in  the  best 
modes  of  presenting  the  subject  to  the  minds  of  his  pupils  in  the 
various  forms  of  practical  instruction. 

The  work  is  systematic  in  its  arrangement ;  it  contains  all  that 
will  be  useful  in  Common  Schools,  and  is  just  what  is  wanted  to 
make  a  thinking  pupil.  We  can,  therefore,  commend  it  to  the  notice 
and  patronage  of  Teachers,  Parents,  and  School  Committees ;  be- 
lieving that  where  it  is  used  the  pupils  will  acquire  not  only  a  com- 
petent knowledge  of  Algebra,  but,  at  the  same  time,  they  will  oe 
making  as  much  progress  in  Arithmetic,  as  they  could,  if  required 
to  give  their  exclusive  attention  to  the  best  text-books  now  used  in 
Oral  Arithmetic. 

ConNELius  Walker,  Richard  G.  Parkbr, 

Samuel  Barrett,  W.  J.   Adams, 

Abner  Forbics,  Frederick  Crafts, 

Charles  B.  Sherman,  Albert  Bowkkr, 

Thomas  Baker,  .Iosiah  A.  Stearns, 

Joshua  Bates,  Jr.,  [saac  F.  Shepard, 

•  iKORnK  B.  Uydk,  Qrammar  Master*. 


CiiARLESTOWN,  July  11, 1845. 
Dear  Sir, — 1  have  the  ])'easure  to  inform  you  that  after  a  careful 
ex  iiniiiatioii  on  the  part  of  our  Board  of  Trustees,  of  your  "'  Intel- 
leciual  Mgebra,'^  it  was  unanimously  voted  lo  introduce  it  into  our 
Grammar  Schools.  Some  of  our  Teachers  have  thoroughly  exam- 
ined the  hook,  and  speak  in  hi^h  terms  of  its  merits. 

Ilespectfully  yours,      JONATHAN  BROWN,  Jk., 

Secretary 
To  D.  B.  Tower,  Esq. 

Mr.  Pierce,  the  experienced  Principal  of  the  Normal  School 
West  Newton,  June  2Gth,  writes,  "  I  am  so  well  pleased  with  i/ 
(the  Algebra),  that  I  propose  to  introduce  it  into  the  JNlodel  Schoof 
next  Term." 

Chelsea,  July  9,  184-5. 
Mr.  Tower, — Dear  Sir :  I  have  e.xamined  your  '■'  Intellectual  Algebra' 
and  [  should  be  much  gratified  at  its  introduction  into  the  School 
under  my  charge.  I  find  the  mental  exercises  in  the  Arithmetic  we 
use  altogether  inadequate,  and  am  confident  that  the  introduction  of 
your  work,  at  this  stage  of  the  scholars  progress,  will  enable  him 
to  understand  the  science  of  Arithmetic  much  better  and  more  easily 
than  he  can  now  do. 

Respectfully,  QUINCY  ADAMS. 

Charlestown,  July  S,  1845. 
Mr.  Towa; — Dear   Sir:  Your  work  on  ^^  Intellectual  jllgebra,"  we 
have  examined  with  much  interest,  and  a  high  degree  of  pleasure 
The  idea  of  the  work  is  excellent,  and  the  ari^angeraent,  we  think 
is  good. 

It  is  the  first  book  of  the  kind  that  we  have  seen,  and  it  appears 
to  be  well  calculated  to  supply  a  deficiency  in  the  class  of  books  for 
the  intellectual  training  of  the  youthful  mind.  A  more  interesting, 
useful,  and  important  work  could  hardly  have  been  devised,  and  it 
cannot  i'ail,  we  think,  to  meet  the  approbation  of  Teachers  and 
friends  of  education. 

Very  respectfully,  P.  H.  SAVEETSER, 

Principal  of  Grammar  Department  of  Haivard  School. 

DANIEL  H.  FORBES,      ' 
PriiKipal  of  Grammar  Departm^.tit  of  Warren  School 

A.  WALKER, 
Principal  of  Grammar  Department  of  mnlhrop  School 

Charlestow.n,  July  19,  iSlo. 
We  have  examined,  carefully  and  with  much  satisfaction,  Tower's 
"■  Intellectual  Algebra,"  which  bears  the  same  relation  to  the  Algebraic 
text-books  in  common  use,  as  that  sustained  by  ''  Colburn's  First 
Lessons"  to  previous  treatises  upon  Arithmetic — and  we  think  that 
every  one,  who  has  made  use  of  that  excellent  work,  cannot  fail  to 
regttrd  this  as  the  highest  commendation.      We  are  highly  gratified 


to  learn  that  the  Tiustees  have  introduced  the  work  into  the  Schools 
under  our  care. 

BENJAMIN  F.TWEED, 
Principal  of  Bunker  Hill  School. 
JOSEPH  T.  SWAN, 
Principal  of  Mathematical  Department  of  Warren  School. 

STACY  BAXTER, 

Principal  of  Mathematical  Department  of  Winthrop  Sci^ol. 

From  Professor  Forbes,  Civil  Engineer,  formerly  Principal  of  the  High 
School  in  Lowell. 

Lowell,  July  21,  1845. 
Dear  Sir — I  have  examined  your  '^  Intellectual  Jlgcbra"  with  in 
terest;  and  I  believe  it  will  be  found  highly  useful  in  giving  to  the 
young  habits  of  thinking  attentively,  and  of  reasoning  with  pre- 
cision— two  of  the  most  desirable  results  of  education.  Your  book 
is  the  best  of  its  kind  that  I  have  seen. 

Very  respectfully  Yours,         FRANKLIN  FORBES. 
David  B.  Tower,  Esq. 

Salem,  July  12,  1845. 
D.  B.  Tower,  Esq. — Dear  Sir :  I  have  examined  with  much  atten- 
tion your  "  Intellectual  Jlls^cbra."  I  think  the  plan  of  the  work  is 
excellent;  and  so  far  'as  I  have  examined,  the  filling  up  is  equally 
good.  I  suspect  you  have  done  for  Algebra  a  service  not  very  unlike 
what  Colburn  did  for  Arithmetic,  when  he  published  his  "i-'iVsf 
Lessons.^^  I  have  requested  our  School  Committee  to  allow  me  to 
nut  it  into  the  hands  of  my  Junior  Class,  as  a  preparatory  study. 
Yours,  very  respectfully,  RUFUS  PUTNAM, 

Principal  of  the  Bowditch  English  High  School,  Salem,  Mass. 

Boston  Daily  Journal. 
The  plan  of  this  work  is  altogether  new — it  contemplates  the 
improvement  in  the  mode  of  teaching  Algebra,  that  Colburn  intro- 

luced  into  Arithmetic  some  twenty  years  ago,  viz. — by  oral  exer- 
cises, in  which  all  the  operations  are  limited  to  such  small  numbers 
as  not  to  embarrass  the  reasoning  powers,  but  on  the  inductive  plan, 
to  lead  the  pupil,  understandingly,  step  by  step,  to  higher  mental 
efforts  *  *  *  *  We  think  its  merits  will  be  found 
lo  entitle  it  to  admission  into  our  schools  as  a  valuable  aid  to  the 

I'eachers  in  giving  instruction  in  Algebra  to  our  youthful  readers. 

Mass.  Temperance  Standard,  .Aug.  1,  1845. 
We  have  looked  over  this  work  with  much  interest.  To  most 
persons,  the  idea  of  the  study  of  Algebra,  is  that  of  a  hard,  dry, 
useless  task  ;  and  formerly  this  idea  was  in  the  main  correct.  Some 
of  the  early  treatises  on  this  subject  seem  to  have  been  intended 
lo  convey  the  little  information  they  contained,  in  as  blind  a  method 
as  possible.  But  Warren  Colburn,  by  his  excellent  treatise,  made 
the  translation  from  the  study  of  iVrithmetic  to  that  of  Algebra,  easy 


and  delightful    Not  content  with  this  advance,  Mr.  Tower  has  now 

prepared  a  treatise,  which  is  designed  to  hold  the  same  position  in 
reference  to  Algebra  that  Mr.  Colbum's  ■'■  Intelkctual  JrithnKtic"  does 
to  Arithmetic — that  is.  to  make  it  ont  of  the  most  elementary  studies 
in  common  schools.  The  idea  seems  to  us  a  good  one.  There  is 
nothing  in  the  nature  of  Algebra  to  render  it  adiliicult  study.  Il 
any  otie  doubts  this  statement,  let  him  read  over  Mr.  Tower's  book, 
and  he  will  be  sceptical  no  longer.  But  what  is  of  still  highei 
importance,  the  child  by  these  steps,  which  seem  so  pleasant  and 
simple,  is  learning  the  greatest  of  all  arts — that  of  reasoning  In 
this  age  of  loose  reasoners,  every  man  who  does  anything  to  direct 
the  minds  of  the  young  to  habit;  of  closer  investigation  and  analysis, 
does  a  service  to  the  community  which  cannot  easily  be  over-rated. 
In  this  respect  it  gives  us  great  pleasure  to  recommend  the  little 
treatise  of  Mr  Tower. 

Boston  Messenqer^  July  31,  IS45. 

"  Litellectual  Algebra ;  or,  Oral  E.xercises  in  Algebiu,  for  Common 
Schools — in  which  all  the  operations  are  limited  to  such  small 
numbers  as  not  to  embarrass  the  reasoning  powers,  but,  on  the  in- 
ductive plan,  to  lead  the  pupil  understandingly,  step  by  step,  to 
higher  mental  efforts ,  adapted  to  prepare  the  pupil  for  the  study  of 
mental  Arithmetic,  and  designed  to  be  introductory  to  higher  treat- 
ises on  Algebra." 

There  is  no  class  of  Works  in  which  the  public  are  more 
aeeply  interested  than  in  School  Books,  and  when  good  ones  are 
published,  the  author  shoulil  be  encouraged,  and  receive  the  com 
mendation  that  his  labors  deserve.  It  is  with  this  feeling  that  we 
ahv.'iys  notice  school  books,  and  in  the  present  instance  we  are 
happy  in  being  able  to  spe  ik  favorably  of  a  valuable  addition  to  our 
stock  of  books,  on  a  most  interesting  and  important  study,  which, 
by  means  of  this  treatise,  may  be  introduced  with  the  greatest  ad 
vantage  into  our  public  schools.  We  will  only  add,  that  the  plan  ol 
the  author  is  admirably  executed. 

The  able  Editor  of  the  Christian  Reflector,  who  was  selected  from 
the  Boston  School  Committee  to  e.xamine  the  Mathematical  Depart- 
ment of  their  Schools,  and  who  has  just  completed  that  arduous 
task,  says  of  Towers  "  IrUdlectwd  Algebra''' — 

'•  This  is  a  new  text-book,  on  a  new  plan,  which  we  greatly  admire. 
It  is  to  the  Algebraic  science  very  much  such  a  work  as  was  Col- 
burn's  '  FiVsf  ^ri/Ame/jc'  to  the  science  of  common  numbers.  We 
observe  that  it  is  commended  by  e.Kperienced  teachers.  We  shall 
tertainly  favor  its  adoption  in  the  Mathematical  department  of  the 
Schools  of  Boston,  and  recommend  it  to  the  attention  of  School 
Committees  throughout  the  countrj-.' 


The  following  is  from  the  Principal  of  the  celebrated  Private 
School  in  Roxbury,  one  of  the  best  in  this  country. 

David  B.  Town;  Esq., — Dear  Sir :  I  have  examined  your  "  Intel 
Iccitial  Algebra"  with  some  care  and  attention,  and  am  much 
pleased  vvith  the  plan  and  execution  of  the  work.  I  think  it 
admirably  adapted  for  the  early  training  of  youthful  minds  in 
mathematics.  I  shall  introduce  it  forthwith  into  my  school. 
Very  truly  and  sincerely  yours, 

DANIEL  LEACH. 
Roxtury,  August,  6,  1845. 

From  K.  G.  Slorke,  Esq.,  County  Superintendent  of  Cai/uga  County 

Auburn,  Sept.  20,  1845. 

Messrs    Paine  &,■  Burgess, — The  examination  of''  Towers'  Intelle' 
lual  Algebra"  led  me  to  remark  that  it  was  a  work  whiclu   I  cout -• 
cheerfully  and  heartily  recommend,  for  its   intrinsic  value  and  ei. 
cellence  ;  and  I  avail  myself  of  the  (irst  opportunity  of  doing  so. 

I  regard  it  as  the  legitimate  successor  of  Colburn's  Fir-t  LessonJ, 
and  it  will,  in  my  opinion,  prove  as  valuable  to  the  student  of 
Algebra  as  that  has  been  to  the  student  of  Arithmetic.  It  divests 
the  science  of  its  mystery  aud  repulsiveness,  and  brings  hs  principlea 
clearly  before  the  mental  vision,  so  sirnplitied  and  illustrated,  that 
they  can  be  readily  comprehended  by  most  pupils  of  from  ten  to 
twelve  years  of  age. 

I  therefore  hail  with  pleasure,  this  new  and  valuable  incentive  to 
mental  exercise  in  our  Schools, and  am  satisfied  that  the  work  has  but 
to  be  examined  to  be  approved  and  adopted.  It  is  peculiarly  adapt- 
ed to  the  use  of  Common  Schools,  and  to  facilitate  its  introduc- 
tion, we  shall  give  the  members  of  our  Teachers'  Institute,  which 
is  sojn  to  convene,  daily  and  thorough  exercises  in  it. 

Respectfully  and  truly  Yours,  E.  G.  STORKE. 

Boston,  Sept.  23,  1S4.5. 

Dear  Sir, — Having  been  absent  from  the  city  several  months,  I  did 
not  leceive,  so  soon  as  I  otherwise  should,  the  copy  of  your  book, 
the  "  Intellectual  Algebra,"  which  you  did  me  the  honor  to  send  to 
my  house.  I  have  examined  the  book  within  a  few  days,  and  in 
my  humble  opinion,  it  is  admirably  adapted  to  the  purposes  foi 
which  it  is  inteniled. 

It  seems  to  me,  you  have  very  happily  applied  the  "  charms  ol 
logic"  to  that  beautiful  and  much  neglected  study  of  Algebra,  and 
if  such  a  book  could  be  freely  introduced  into  our  Common  Schools 
I  doubt  not  it  would  do  more  than  almost  anything  else  to  invigo 
rate  and  concentrate  the  intellectual  powers  of  the  young. 
With  much  respect,  your  obliged  servant, 

JOHN   T.  SARGENT 

Di»viD  B.  Tower,  Esq. 


Salem,  July  26,  1845.  ' 

JIfr.  Darid  B.  Ibwer, — Dear  Sir:  It  is  thought  by  most  Teachers 
at  present,  that  children  have  not  commenced  the  study  of  Arith- 
metic aright  and  radically,  unless  they  have  begun  with  "  Colburn's 
First  Lessons,"  or  some  other  book  of  oral  exercises.  It  appears  to 
us  that  it  is  equally  important  that  Jllgcbra  should  be  thus  com- 
menced. We  rejoice  to  see  a  work  of  this  kind  from  your  hands; 
and  the  wonder  is,  that  it  has  not  entered  the  brain  of  some  one 
before,  to  put  one  forth.  Your  ^'Intellectual  Algebra"  in  our  humble 
opinion,  is  a  happy  conception,  and  a  design  well  executed, — leading 
tlie  mind  on  by  very  easy  and  gradual  steps,  and  by  clear  illustra- 
tions. We  regard  Algebra  as  an  interesting  and  important  study 
for  children,  and  well  calculated  to  aid  their  progress  in  cornmon 
Arithmetic.  We  think,  that  if  the  merits  of  the  study,  and  of  youi 
little  book,  are  duly  appreciated,  it  will  be  widely  introduced  into 
fhe  Schools  of  our  land. 

Yours  with  esteem,         EDWIN  JOCELYN, 

Prindjtal  of  F.  High  School. 

CHARLES  NORTHEND, 

Principal  of  Epes  School 

D.  P..GALLOUP, 

Principal  of  Hacker  School 

A    C.  SxMITH, 

Pririnpal  of  Pliilip's  School 
,.  .^.I.  B.  FAIRFIELD, 

Principal  of  Browne  School 


From  Boston  Itcconicr,  July  31,  1S45. 

This  work  was  prepared,  the  author  informs  us,  for  the  use  of 
the  blind  under  his  charge,  and  is  now  printed  in  hope  that  it  may 
prove  useful  to  the  seeing.  It  is  on  the  "  inductive  plan,"  and  is 
believed  to  supply  a  deficiency  in  the  books  provided  for  young 
pupils.  The  operations  are  limited  to  small  numbers,  and  lead  the 
pupil  on  step  by  step  towards  higher  mental  efforts.  The  jtlan, 
and  the  execution  of  it,  cannot  fail  to  meet  the  approbation  of 
Teachers. 

Boston,  Skpt.,  15,  1S1.5. 
D.  J>.    Totrvr.  J'^q., — Bear  Sir: — I  have  examined  your   "■Intel- 
lectiial  Algihra."  and  cheerfully  concur  in  the  opinion  expressed  in 
the  n-commendalion  of  the  Principals  of  the  Public  Schools  in 
Boston.  Veiy  respectfully  yours, 

R.  W.  WRIGHT, 
Fnncipal  of  the  department  of  Mathematics  in  the  Jldams  School 


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S.I  »h»s  Quar(9,  or  Second.  Book  in  «eoe-»yhy. 
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rmotre  (A»las  h  ■    ,<  jt  viifl-  c-^vvs). 
Sir    '^iS  ][  .i<  ■»<>  A^ilhnieti' 

5       i'.-v  :"-    p(ic<4<  and  ITlental  Aritbinetic  and  Key. 
t«inl:iOs(  Nenr  Aritbmetic  and  Key. 

BT  ASA  SMITH. 
^mitli'f>.  Illustrated  Astronomy. 
Muiia^s  Abrid^<-d  «' 

.  "mith's   Iliuslritted  «  In  Spanish  Language. 

D.  B.  TOWEB'S  SEEIES. 
To  trerN  Speller  and  Complete  Enunciator. 
Tower's  First  Reader,  or.  Gradual  rrimer,  Enlarged  and  Illnstrate'd, 
Totwer's  Second  Reader,  or,  Introduction  to  Gtadnul   Reader,  Eul 

f.nd  Illustnted. 
Towor'x  'titird  Render,  or,  Grudua)  Rearior,  Enlarged. 
'''•-»werN  Fourtli   Header,  or,  y.-qt>el  to  the  GiaduHl  Read.r. 
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Towel's  Sixtli  Reader,  or,  N.  A.  First  Oass  Re:  l.r. 
Tjwrer'w  Clemc'    s  of  '"    •mm;-"- 
r      (T*  's  OramivKr  aii  i  X         »•». 
Tivc;  J  Alffe*».        •  >d- W«  s 

Hd*     fr;:ie        i    md  Fr   ,..h   Irithmi-tics. 
*•  -*•  .»»..  I's  :\'atural  Hii«.tor^. 
'■»i«;.srt'ii'»  Expositor  and  Eioentieru  made  Fa»y. 

-u.rnsey's  ITnited  states,  Juvenile  and  Advanced. 
-Mart's  Geoc-raphieal  Uucslions  and  Atlas.   . 
Mayliew's  Praclicat  Book-keeping:  and  Blanks, 
'la'     eT»'»«  Key  to  Do. 
.May  «e»v  on  Popular  Education. 
^    lUb ban's  Speller,  Dcfiner,  and  Reader,  In  Two  Paris. 

;    >r-rd's  Youth's  GrainmaT. 


mm 


'fW 


^  atever's  itfodern  R-ildev's  Guide. 
^>uld*s  nonBr-rurt...r«ter's  and  Joiner^s  Assistant. 
I'lans  for  C'bnrcStes.     V>^li\  20  r)«-signs  hy  Eminent  Arrhltecu. 


